| L(s) = 1 | + 2·3-s + 4-s − 2·7-s + 9-s + 2·12-s + 8·13-s + 16-s + 4·19-s − 4·21-s − 4·27-s − 2·28-s − 8·31-s + 36-s − 4·37-s + 16·39-s − 16·43-s + 2·48-s + 3·49-s + 8·52-s + 8·57-s + 16·61-s − 2·63-s + 64-s + 8·67-s − 4·73-s + 4·76-s + 16·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s + 0.577·12-s + 2.21·13-s + 1/4·16-s + 0.917·19-s − 0.872·21-s − 0.769·27-s − 0.377·28-s − 1.43·31-s + 1/6·36-s − 0.657·37-s + 2.56·39-s − 2.43·43-s + 0.288·48-s + 3/7·49-s + 1.10·52-s + 1.05·57-s + 2.04·61-s − 0.251·63-s + 1/8·64-s + 0.977·67-s − 0.468·73-s + 0.458·76-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.639040694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.639040694\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.190576101673579459695630969447, −7.71895664384636679391588711627, −7.20568692412223659760548448204, −6.66965130292041403123447657278, −6.56614507384142030967141653825, −5.76860542213636603043499309352, −5.59857578974307577358363282055, −4.99929117585498161544137242896, −4.11500666640753372808858013263, −3.62404662713201803361092631083, −3.28327332646103144207952017478, −3.16593838634617734875675320632, −2.07204529193989626181377807359, −1.80953354791836556352638476370, −0.814782882121918462041385743038,
0.814782882121918462041385743038, 1.80953354791836556352638476370, 2.07204529193989626181377807359, 3.16593838634617734875675320632, 3.28327332646103144207952017478, 3.62404662713201803361092631083, 4.11500666640753372808858013263, 4.99929117585498161544137242896, 5.59857578974307577358363282055, 5.76860542213636603043499309352, 6.56614507384142030967141653825, 6.66965130292041403123447657278, 7.20568692412223659760548448204, 7.71895664384636679391588711627, 8.190576101673579459695630969447
