L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 4·13-s − 14-s + 16-s − 2·17-s − 8·19-s + 8·23-s − 4·26-s − 28-s − 8·29-s + 4·31-s + 32-s − 2·34-s + 8·37-s − 8·38-s − 12·41-s − 8·43-s + 8·46-s − 4·47-s + 49-s − 4·52-s − 6·53-s − 56-s − 8·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 1.83·19-s + 1.66·23-s − 0.784·26-s − 0.188·28-s − 1.48·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s + 1.31·37-s − 1.29·38-s − 1.87·41-s − 1.21·43-s + 1.17·46-s − 0.583·47-s + 1/7·49-s − 0.554·52-s − 0.824·53-s − 0.133·56-s − 1.05·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.245799437086343760393736146379, −7.37809344928501113201345267535, −6.66222910486457306007719363045, −6.13948866156581164180828120483, −4.98097563320052319862799855392, −4.60504583712161652651498845302, −3.52477719735270810583168107695, −2.69330686345084354832872734018, −1.77998975598385783353163081350, 0,
1.77998975598385783353163081350, 2.69330686345084354832872734018, 3.52477719735270810583168107695, 4.60504583712161652651498845302, 4.98097563320052319862799855392, 6.13948866156581164180828120483, 6.66222910486457306007719363045, 7.37809344928501113201345267535, 8.245799437086343760393736146379