| L(s) = 1 | − 1.45·2-s + 2.41·3-s + 0.110·4-s − 3.50·6-s − 1.25·7-s + 2.74·8-s + 2.82·9-s + 2.46·11-s + 0.266·12-s + 2.06·13-s + 1.81·14-s − 4.20·16-s − 1.42·17-s − 4.09·18-s − 0.479·19-s − 3.01·21-s − 3.58·22-s − 2.87·23-s + 6.62·24-s − 3.00·26-s − 0.433·27-s − 0.137·28-s − 9.26·29-s + 0.180·31-s + 0.623·32-s + 5.94·33-s + 2.06·34-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 1.39·3-s + 0.0551·4-s − 1.43·6-s − 0.472·7-s + 0.970·8-s + 0.940·9-s + 0.743·11-s + 0.0767·12-s + 0.574·13-s + 0.485·14-s − 1.05·16-s − 0.345·17-s − 0.965·18-s − 0.110·19-s − 0.658·21-s − 0.763·22-s − 0.599·23-s + 1.35·24-s − 0.589·26-s − 0.0833·27-s − 0.0260·28-s − 1.72·29-s + 0.0324·31-s + 0.110·32-s + 1.03·33-s + 0.354·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 3 | \( 1 - 2.41T + 3T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 - 2.46T + 11T^{2} \) |
| 13 | \( 1 - 2.06T + 13T^{2} \) |
| 17 | \( 1 + 1.42T + 17T^{2} \) |
| 19 | \( 1 + 0.479T + 19T^{2} \) |
| 23 | \( 1 + 2.87T + 23T^{2} \) |
| 29 | \( 1 + 9.26T + 29T^{2} \) |
| 31 | \( 1 - 0.180T + 31T^{2} \) |
| 37 | \( 1 - 0.824T + 37T^{2} \) |
| 41 | \( 1 - 1.12T + 41T^{2} \) |
| 43 | \( 1 + 3.43T + 43T^{2} \) |
| 47 | \( 1 + 3.00T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 - 0.433T + 59T^{2} \) |
| 61 | \( 1 + 3.88T + 61T^{2} \) |
| 67 | \( 1 - 5.24T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 4.35T + 73T^{2} \) |
| 79 | \( 1 + 7.28T + 79T^{2} \) |
| 83 | \( 1 + 1.09T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 9.80T + 97T^{2} \) |
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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
−8.129094768925847374486846047280, −7.28971617051867657719777216864, −6.64048857424367422517888232221, −5.73038488357579133937931155540, −4.53474417015480912203847613925, −3.83099581928181860096635190303, −3.22252219246109890428807569663, −2.08506366592910514684656520073, −1.44747044836572282743460766097, 0,
1.44747044836572282743460766097, 2.08506366592910514684656520073, 3.22252219246109890428807569663, 3.83099581928181860096635190303, 4.53474417015480912203847613925, 5.73038488357579133937931155540, 6.64048857424367422517888232221, 7.28971617051867657719777216864, 8.129094768925847374486846047280
