| L(s) = 1 | − 3-s − 3·5-s − 3·7-s − 9-s − 4·11-s − 2·13-s + 3·15-s + 3·17-s − 12·19-s + 3·21-s + 25-s − 6·31-s + 4·33-s + 9·35-s − 7·37-s + 2·39-s + 6·41-s − 3·43-s + 3·45-s + 9·47-s − 3·49-s − 3·51-s − 18·53-s + 12·55-s + 12·57-s + 12·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 1.13·7-s − 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.774·15-s + 0.727·17-s − 2.75·19-s + 0.654·21-s + 1/5·25-s − 1.07·31-s + 0.696·33-s + 1.52·35-s − 1.15·37-s + 0.320·39-s + 0.937·41-s − 0.457·43-s + 0.447·45-s + 1.31·47-s − 3/7·49-s − 0.420·51-s − 2.47·53-s + 1.61·55-s + 1.58·57-s + 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T - 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 76 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 170 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 2 T - 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 124 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 38 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.92935982090090269524877935962, −10.67565088589767443246463949915, −10.14961185105258290925313438877, −9.817551941800209256085620937533, −9.079843229105532359254548938110, −8.686642160719022342144432527557, −8.113845820974984917274798486075, −7.80661199171213693905190107623, −7.25185454608095247488917996700, −6.80989501401375884306401767831, −6.16947279325834065838914490210, −5.81669187073265291085738243765, −5.19032220429870854510795163877, −4.55268808714137228837513162197, −4.00928377650971953596887185380, −3.45306201677728978971783322968, −2.80712463094253039902674874006, −2.02601789412831845206354750598, 0, 0,
2.02601789412831845206354750598, 2.80712463094253039902674874006, 3.45306201677728978971783322968, 4.00928377650971953596887185380, 4.55268808714137228837513162197, 5.19032220429870854510795163877, 5.81669187073265291085738243765, 6.16947279325834065838914490210, 6.80989501401375884306401767831, 7.25185454608095247488917996700, 7.80661199171213693905190107623, 8.113845820974984917274798486075, 8.686642160719022342144432527557, 9.079843229105532359254548938110, 9.817551941800209256085620937533, 10.14961185105258290925313438877, 10.67565088589767443246463949915, 10.92935982090090269524877935962
