| L(s) = 1 | − 2-s − 1.05·3-s + 4-s − 2.13·5-s + 1.05·6-s + 3.42·7-s − 8-s − 1.88·9-s + 2.13·10-s − 2.88·11-s − 1.05·12-s + 1.88·13-s − 3.42·14-s + 2.26·15-s + 16-s + 4.09·17-s + 1.88·18-s − 3.98·19-s − 2.13·20-s − 3.62·21-s + 2.88·22-s + 23-s + 1.05·24-s − 0.428·25-s − 1.88·26-s + 5.16·27-s + 3.42·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.610·3-s + 0.5·4-s − 0.956·5-s + 0.431·6-s + 1.29·7-s − 0.353·8-s − 0.627·9-s + 0.676·10-s − 0.871·11-s − 0.305·12-s + 0.522·13-s − 0.915·14-s + 0.583·15-s + 0.250·16-s + 0.993·17-s + 0.443·18-s − 0.915·19-s − 0.478·20-s − 0.790·21-s + 0.615·22-s + 0.208·23-s + 0.215·24-s − 0.0856·25-s − 0.369·26-s + 0.993·27-s + 0.647·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 1.05T + 3T^{2} \) |
| 5 | \( 1 + 2.13T + 5T^{2} \) |
| 7 | \( 1 - 3.42T + 7T^{2} \) |
| 11 | \( 1 + 2.88T + 11T^{2} \) |
| 13 | \( 1 - 1.88T + 13T^{2} \) |
| 17 | \( 1 - 4.09T + 17T^{2} \) |
| 19 | \( 1 + 3.98T + 19T^{2} \) |
| 29 | \( 1 - 7.47T + 29T^{2} \) |
| 31 | \( 1 + 8.99T + 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 - 8.18T + 41T^{2} \) |
| 43 | \( 1 - 0.248T + 43T^{2} \) |
| 47 | \( 1 + 6.96T + 47T^{2} \) |
| 53 | \( 1 - 6.38T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 - 9.84T + 71T^{2} \) |
| 73 | \( 1 - 2.29T + 73T^{2} \) |
| 79 | \( 1 - 0.874T + 79T^{2} \) |
| 83 | \( 1 + 9.68T + 83T^{2} \) |
| 89 | \( 1 + 0.610T + 89T^{2} \) |
| 97 | \( 1 - 5.60T + 97T^{2} \) |
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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
−7.77023402022725498659202808100, −7.32596514780035934138599162661, −6.32311908294611297497022185281, −5.54119491776899447979682477226, −4.99547467928131889233615760697, −4.08744683460138472351676023144, −3.15802985619715790107922631478, −2.14885845714706907506718338076, −1.04838368086083808633468380519, 0,
1.04838368086083808633468380519, 2.14885845714706907506718338076, 3.15802985619715790107922631478, 4.08744683460138472351676023144, 4.99547467928131889233615760697, 5.54119491776899447979682477226, 6.32311908294611297497022185281, 7.32596514780035934138599162661, 7.77023402022725498659202808100
