L(s) = 1 | + 0.856·2-s − 1.26·4-s + 0.647·7-s − 2.79·8-s − 5.91·11-s + 2.88·13-s + 0.555·14-s + 0.134·16-s + 4.20·17-s − 1.64·19-s − 5.06·22-s + 6.42·23-s + 2.47·26-s − 0.820·28-s + 2.25·29-s − 5.28·31-s + 5.71·32-s + 3.59·34-s − 1.71·37-s − 1.41·38-s − 0.176·41-s + 7.95·43-s + 7.48·44-s + 5.50·46-s + 1.05·47-s − 6.58·49-s − 3.65·52-s + ⋯ |
L(s) = 1 | + 0.605·2-s − 0.632·4-s + 0.244·7-s − 0.989·8-s − 1.78·11-s + 0.799·13-s + 0.148·14-s + 0.0336·16-s + 1.01·17-s − 0.378·19-s − 1.08·22-s + 1.34·23-s + 0.484·26-s − 0.155·28-s + 0.419·29-s − 0.949·31-s + 1.00·32-s + 0.617·34-s − 0.282·37-s − 0.229·38-s − 0.0275·41-s + 1.21·43-s + 1.12·44-s + 0.811·46-s + 0.154·47-s − 0.940·49-s − 0.506·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 0.856T + 2T^{2} \) |
| 7 | \( 1 - 0.647T + 7T^{2} \) |
| 11 | \( 1 + 5.91T + 11T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 - 4.20T + 17T^{2} \) |
| 19 | \( 1 + 1.64T + 19T^{2} \) |
| 23 | \( 1 - 6.42T + 23T^{2} \) |
| 29 | \( 1 - 2.25T + 29T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 37 | \( 1 + 1.71T + 37T^{2} \) |
| 41 | \( 1 + 0.176T + 41T^{2} \) |
| 43 | \( 1 - 7.95T + 43T^{2} \) |
| 47 | \( 1 - 1.05T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 - 9.74T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 + 6.09T + 71T^{2} \) |
| 73 | \( 1 + 7.08T + 73T^{2} \) |
| 79 | \( 1 - 1.58T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 7.55T + 97T^{2} \) |
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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
−7.86664446689468208894492084248, −7.12004578006130999876934135608, −6.03537812253530997130609741139, −5.48428189493884168214326271892, −4.95346271184747348427411490353, −4.18104473614669236607045371142, −3.23768436832519086944858537549, −2.70939383462823917588567568602, −1.28771562876727834514880912486, 0,
1.28771562876727834514880912486, 2.70939383462823917588567568602, 3.23768436832519086944858537549, 4.18104473614669236607045371142, 4.95346271184747348427411490353, 5.48428189493884168214326271892, 6.03537812253530997130609741139, 7.12004578006130999876934135608, 7.86664446689468208894492084248