| L(s) = 1 | + 2·2-s + 3·4-s − 7-s + 4·8-s − 2·9-s − 2·14-s + 5·16-s − 4·18-s − 3·28-s − 12·29-s + 6·32-s − 6·36-s − 4·37-s − 16·43-s + 49-s − 12·53-s − 4·56-s − 24·58-s + 2·63-s + 7·64-s + 8·67-s − 8·72-s − 8·74-s + 16·79-s − 5·81-s − 32·86-s + 2·98-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.377·7-s + 1.41·8-s − 2/3·9-s − 0.534·14-s + 5/4·16-s − 0.942·18-s − 0.566·28-s − 2.22·29-s + 1.06·32-s − 36-s − 0.657·37-s − 2.43·43-s + 1/7·49-s − 1.64·53-s − 0.534·56-s − 3.15·58-s + 0.251·63-s + 7/8·64-s + 0.977·67-s − 0.942·72-s − 0.929·74-s + 1.80·79-s − 5/9·81-s − 3.45·86-s + 0.202·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.71895664384636679391588711627, −7.60556812848957435561623382149, −6.73571587686584923478967639363, −6.66965130292041403123447657278, −6.15190325448345686265108903037, −5.59857578974307577358363282055, −5.12614454716073682582164065811, −5.04607295657079957887442801881, −4.11500666640753372808858013263, −3.77049860941673777607586608067, −3.28327332646103144207952017478, −2.83590121110111537782644859852, −2.07204529193989626181377807359, −1.54150783577442522625509630661, 0,
1.54150783577442522625509630661, 2.07204529193989626181377807359, 2.83590121110111537782644859852, 3.28327332646103144207952017478, 3.77049860941673777607586608067, 4.11500666640753372808858013263, 5.04607295657079957887442801881, 5.12614454716073682582164065811, 5.59857578974307577358363282055, 6.15190325448345686265108903037, 6.66965130292041403123447657278, 6.73571587686584923478967639363, 7.60556812848957435561623382149, 7.71895664384636679391588711627
