| L(s) = 1 | + 2·4-s − 2·9-s − 8·11-s + 3·16-s − 8·29-s − 16·31-s − 4·36-s + 32·41-s − 16·44-s + 12·49-s − 8·59-s + 8·61-s + 12·64-s + 8·71-s + 3·81-s − 48·89-s + 16·99-s + 8·101-s + 24·109-s − 16·116-s − 4·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s − 6·144-s + ⋯ |
| L(s) = 1 | + 4-s − 2/3·9-s − 2.41·11-s + 3/4·16-s − 1.48·29-s − 2.87·31-s − 2/3·36-s + 4.99·41-s − 2.41·44-s + 12/7·49-s − 1.04·59-s + 1.02·61-s + 3/2·64-s + 0.949·71-s + 1/3·81-s − 5.08·89-s + 1.60·99-s + 0.796·101-s + 2.29·109-s − 1.48·116-s − 0.363·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/2·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.370252621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.370252621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 17 | $C_4\times C_2$ | \( 1 + 4 T^{2} + 70 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3670 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 20566 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 12134 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 30166 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 24 T + 314 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 316 T^{2} + 43270 T^{4} - 316 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.36256996510191464630543754572, −7.01345150121244380997268399794, −6.77064957575023757283151500966, −6.57806081979420055479487990172, −6.18050745953012928292312499152, −5.79333312753433223489266846666, −5.76812870874958251909192061529, −5.59148833463464964694904636914, −5.47253410739886357276672258338, −5.40731166354287391844085444498, −4.91034121126452710860501708403, −4.50108332377588950532099191872, −4.49224849754064866300322219559, −3.98509939943295872239769294276, −3.69591931518558975956657042016, −3.64130468942501621101632294348, −3.33549921202542501754483141298, −2.60186256966660128009654903175, −2.56134256279391386554744594220, −2.49655701525501391356363088685, −2.48329095870409955935573450255, −1.70230917544530293627244927122, −1.54404130552665583382355219566, −0.839522067862243753089189013516, −0.29447920915469462073745020250,
0.29447920915469462073745020250, 0.839522067862243753089189013516, 1.54404130552665583382355219566, 1.70230917544530293627244927122, 2.48329095870409955935573450255, 2.49655701525501391356363088685, 2.56134256279391386554744594220, 2.60186256966660128009654903175, 3.33549921202542501754483141298, 3.64130468942501621101632294348, 3.69591931518558975956657042016, 3.98509939943295872239769294276, 4.49224849754064866300322219559, 4.50108332377588950532099191872, 4.91034121126452710860501708403, 5.40731166354287391844085444498, 5.47253410739886357276672258338, 5.59148833463464964694904636914, 5.76812870874958251909192061529, 5.79333312753433223489266846666, 6.18050745953012928292312499152, 6.57806081979420055479487990172, 6.77064957575023757283151500966, 7.01345150121244380997268399794, 7.36256996510191464630543754572
