| L(s) = 1 | − 2·2-s + 3·4-s + 5·7-s − 4·8-s − 5·9-s + 2·11-s − 10·14-s + 5·16-s + 10·18-s − 4·22-s − 12·23-s + 25-s + 15·28-s − 6·29-s − 6·32-s − 15·36-s − 14·37-s + 16·43-s + 6·44-s + 24·46-s + 18·49-s − 2·50-s − 6·53-s − 20·56-s + 12·58-s − 25·63-s + 7·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.88·7-s − 1.41·8-s − 5/3·9-s + 0.603·11-s − 2.67·14-s + 5/4·16-s + 2.35·18-s − 0.852·22-s − 2.50·23-s + 1/5·25-s + 2.83·28-s − 1.11·29-s − 1.06·32-s − 5/2·36-s − 2.30·37-s + 2.43·43-s + 0.904·44-s + 3.53·46-s + 18/7·49-s − 0.282·50-s − 0.824·53-s − 2.67·56-s + 1.57·58-s − 3.14·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8307506751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8307506751\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−8.285662434899332908926558240119, −8.283254682681969568742412941640, −7.56981196469027501153188760806, −7.44617088698970221673072177182, −6.75549642500640203917529221375, −6.07909319757312535714712792824, −5.73876439085829035805370640147, −5.44015915433775770833668591547, −4.72556975951210881700191573231, −4.01173135747517138761270457828, −3.53795176024353637912410403117, −2.64632677110737282548016864035, −1.97170842587956421114470880781, −1.74843844460655954658244071452, −0.57168783779040589515932752100,
0.57168783779040589515932752100, 1.74843844460655954658244071452, 1.97170842587956421114470880781, 2.64632677110737282548016864035, 3.53795176024353637912410403117, 4.01173135747517138761270457828, 4.72556975951210881700191573231, 5.44015915433775770833668591547, 5.73876439085829035805370640147, 6.07909319757312535714712792824, 6.75549642500640203917529221375, 7.44617088698970221673072177182, 7.56981196469027501153188760806, 8.283254682681969568742412941640, 8.285662434899332908926558240119
