| L(s) = 1 | − 2-s + 1.26·3-s + 4-s − 3.04·5-s − 1.26·6-s + 7-s − 8-s − 1.40·9-s + 3.04·10-s + 3.80·11-s + 1.26·12-s + 5.43·13-s − 14-s − 3.84·15-s + 16-s − 4.12·17-s + 1.40·18-s − 7.58·19-s − 3.04·20-s + 1.26·21-s − 3.80·22-s + 2.48·23-s − 1.26·24-s + 4.25·25-s − 5.43·26-s − 5.56·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.729·3-s + 0.5·4-s − 1.36·5-s − 0.515·6-s + 0.377·7-s − 0.353·8-s − 0.468·9-s + 0.961·10-s + 1.14·11-s + 0.364·12-s + 1.50·13-s − 0.267·14-s − 0.991·15-s + 0.250·16-s − 1.00·17-s + 0.331·18-s − 1.74·19-s − 0.680·20-s + 0.275·21-s − 0.812·22-s + 0.518·23-s − 0.257·24-s + 0.850·25-s − 1.06·26-s − 1.07·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 1.26T + 3T^{2} \) |
| 5 | \( 1 + 3.04T + 5T^{2} \) |
| 11 | \( 1 - 3.80T + 11T^{2} \) |
| 13 | \( 1 - 5.43T + 13T^{2} \) |
| 17 | \( 1 + 4.12T + 17T^{2} \) |
| 19 | \( 1 + 7.58T + 19T^{2} \) |
| 23 | \( 1 - 2.48T + 23T^{2} \) |
| 29 | \( 1 + 9.97T + 29T^{2} \) |
| 31 | \( 1 - 9.59T + 31T^{2} \) |
| 37 | \( 1 - 5.29T + 37T^{2} \) |
| 41 | \( 1 - 2.50T + 41T^{2} \) |
| 43 | \( 1 + 3.18T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 2.66T + 53T^{2} \) |
| 59 | \( 1 - 3.57T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 3.98T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 1.78T + 79T^{2} \) |
| 83 | \( 1 + 4.37T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.953177984363584608624529648467, −7.25773151875630943184234089268, −6.43082298360349569341939494074, −5.90438425609646222647680742822, −4.32926379504672757175050960593, −4.07394409233129652093700318954, −3.23246797005040730558190744550, −2.26905532281643403366611945369, −1.25242675097787894355519677963, 0,
1.25242675097787894355519677963, 2.26905532281643403366611945369, 3.23246797005040730558190744550, 4.07394409233129652093700318954, 4.32926379504672757175050960593, 5.90438425609646222647680742822, 6.43082298360349569341939494074, 7.25773151875630943184234089268, 7.953177984363584608624529648467
