| L(s) = 1 | + 2-s + 4-s + 8-s − 2·9-s + 8·13-s + 16-s − 12·17-s − 2·18-s + 8·26-s − 12·29-s + 32-s − 12·34-s − 2·36-s − 4·37-s + 12·41-s + 49-s + 8·52-s − 12·53-s − 12·58-s + 16·61-s + 64-s − 12·68-s − 2·72-s − 4·73-s − 4·74-s − 5·81-s + 12·82-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2/3·9-s + 2.21·13-s + 1/4·16-s − 2.91·17-s − 0.471·18-s + 1.56·26-s − 2.22·29-s + 0.176·32-s − 2.05·34-s − 1/3·36-s − 0.657·37-s + 1.87·41-s + 1/7·49-s + 1.10·52-s − 1.64·53-s − 1.57·58-s + 2.04·61-s + 1/8·64-s − 1.45·68-s − 0.235·72-s − 0.468·73-s − 0.464·74-s − 5/9·81-s + 1.32·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−7.71895664384636679391588711627, −7.60707532453471338039224955289, −6.66965130292041403123447657278, −6.60348506563741211247253098956, −6.19185902002877876721827038865, −5.59857578974307577358363282055, −5.38990980084792703269563412867, −4.58557293909771964767836479942, −4.11500666640753372808858013263, −3.83664653997120963252560991499, −3.28327332646103144207952017478, −2.50419401332512614244886579026, −2.07204529193989626181377807359, −1.31920416816215806650160602271, 0,
1.31920416816215806650160602271, 2.07204529193989626181377807359, 2.50419401332512614244886579026, 3.28327332646103144207952017478, 3.83664653997120963252560991499, 4.11500666640753372808858013263, 4.58557293909771964767836479942, 5.38990980084792703269563412867, 5.59857578974307577358363282055, 6.19185902002877876721827038865, 6.60348506563741211247253098956, 6.66965130292041403123447657278, 7.60707532453471338039224955289, 7.71895664384636679391588711627
