| L(s) = 1 | − 2-s − 3·5-s − 2·7-s + 8-s + 3·10-s − 8·11-s − 3·13-s + 2·14-s − 16-s − 2·17-s − 19-s + 8·22-s + 23-s + 5·25-s + 3·26-s − 12·31-s + 2·34-s + 6·35-s + 8·37-s + 38-s − 3·40-s + 4·41-s + 6·43-s − 46-s − 2·47-s + 3·49-s − 5·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·5-s − 0.755·7-s + 0.353·8-s + 0.948·10-s − 2.41·11-s − 0.832·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s − 0.229·19-s + 1.70·22-s + 0.208·23-s + 25-s + 0.588·26-s − 2.15·31-s + 0.342·34-s + 1.01·35-s + 1.31·37-s + 0.162·38-s − 0.474·40-s + 0.624·41-s + 0.914·43-s − 0.147·46-s − 0.291·47-s + 3/7·49-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1769026006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1769026006\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 4 T - 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 13 T + 110 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 7 T - 22 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4 T - 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−9.133970303705775399907781801249, −8.703384850718296031286339079443, −8.380985307150271879563876966087, −7.903842742194320840529669973278, −7.65775925697095361332967329054, −7.47686521122505276793023086036, −6.98274740298103474054795386629, −6.78065127179588594301743962305, −5.87346395470072600320149432101, −5.71827095948449913959230065037, −5.22089256028457055065771119046, −4.72005707002373851847232298019, −4.37195260222924538857240390963, −3.96268734758562281318611646834, −3.18348530336096331408348201154, −3.10850891205835588942920262761, −2.23421797727027978624717093921, −2.17906795911070252714031494425, −0.814600805322982599120281158000, −0.21866548320714900126141631064,
0.21866548320714900126141631064, 0.814600805322982599120281158000, 2.17906795911070252714031494425, 2.23421797727027978624717093921, 3.10850891205835588942920262761, 3.18348530336096331408348201154, 3.96268734758562281318611646834, 4.37195260222924538857240390963, 4.72005707002373851847232298019, 5.22089256028457055065771119046, 5.71827095948449913959230065037, 5.87346395470072600320149432101, 6.78065127179588594301743962305, 6.98274740298103474054795386629, 7.47686521122505276793023086036, 7.65775925697095361332967329054, 7.903842742194320840529669973278, 8.380985307150271879563876966087, 8.703384850718296031286339079443, 9.133970303705775399907781801249
