L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 5·11-s − 3·13-s + 14-s + 16-s + 5·17-s − 20-s − 5·22-s − 7·23-s + 25-s + 3·26-s − 28-s − 3·29-s − 6·31-s − 32-s − 5·34-s + 35-s + 2·37-s + 40-s + 41-s + 6·43-s + 5·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 1.50·11-s − 0.832·13-s + 0.267·14-s + 1/4·16-s + 1.21·17-s − 0.223·20-s − 1.06·22-s − 1.45·23-s + 1/5·25-s + 0.588·26-s − 0.188·28-s − 0.557·29-s − 1.07·31-s − 0.176·32-s − 0.857·34-s + 0.169·35-s + 0.328·37-s + 0.158·40-s + 0.156·41-s + 0.914·43-s + 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−12.97745316116617, −12.42146698799283, −12.28544973951840, −11.75245472462486, −11.34922950297279, −10.87928393739384, −10.20144887173544, −9.852543648242180, −9.447081064142923, −9.032638691856154, −8.570086504961764, −7.798098031397960, −7.606696914554328, −7.180502405076705, −6.525793206097843, −6.009843739465960, −5.743838403574130, −4.879982372650098, −4.318223205627133, −3.728613414515680, −3.365196555415279, −2.709739353490419, −1.889661263671838, −1.499738296615964, −0.6886786138371871, 0,
0.6886786138371871, 1.499738296615964, 1.889661263671838, 2.709739353490419, 3.365196555415279, 3.728613414515680, 4.318223205627133, 4.879982372650098, 5.743838403574130, 6.009843739465960, 6.525793206097843, 7.180502405076705, 7.606696914554328, 7.798098031397960, 8.570086504961764, 9.032638691856154, 9.447081064142923, 9.852543648242180, 10.20144887173544, 10.87928393739384, 11.34922950297279, 11.75245472462486, 12.28544973951840, 12.42146698799283, 12.97745316116617