| L(s) = 1 | + 1.09·2-s + 3-s − 0.793·4-s − 3.51·5-s + 1.09·6-s − 3.06·8-s + 9-s − 3.85·10-s − 1.19·11-s − 0.793·12-s + 2.03·13-s − 3.51·15-s − 1.78·16-s + 0.865·17-s + 1.09·18-s − 3.10·19-s + 2.78·20-s − 1.31·22-s − 0.108·23-s − 3.06·24-s + 7.33·25-s + 2.23·26-s + 27-s − 7.73·29-s − 3.85·30-s − 1.05·31-s + 4.17·32-s + ⋯ |
| L(s) = 1 | + 0.776·2-s + 0.577·3-s − 0.396·4-s − 1.57·5-s + 0.448·6-s − 1.08·8-s + 0.333·9-s − 1.21·10-s − 0.361·11-s − 0.228·12-s + 0.565·13-s − 0.906·15-s − 0.446·16-s + 0.209·17-s + 0.258·18-s − 0.712·19-s + 0.622·20-s − 0.280·22-s − 0.0226·23-s − 0.626·24-s + 1.46·25-s + 0.439·26-s + 0.192·27-s − 1.43·29-s − 0.704·30-s − 0.190·31-s + 0.738·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.453088868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.453088868\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 1.09T + 2T^{2} \) |
| 5 | \( 1 + 3.51T + 5T^{2} \) |
| 11 | \( 1 + 1.19T + 11T^{2} \) |
| 13 | \( 1 - 2.03T + 13T^{2} \) |
| 17 | \( 1 - 0.865T + 17T^{2} \) |
| 19 | \( 1 + 3.10T + 19T^{2} \) |
| 23 | \( 1 + 0.108T + 23T^{2} \) |
| 29 | \( 1 + 7.73T + 29T^{2} \) |
| 31 | \( 1 + 1.05T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 43 | \( 1 + 0.138T + 43T^{2} \) |
| 47 | \( 1 + 0.686T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 5.98T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 4.10T + 73T^{2} \) |
| 79 | \( 1 - 1.02T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 + 7.26T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−8.146957276347753514201627006105, −7.41425636417035830373388925371, −6.76583759083629767409026960561, −5.73741301530147015786759785914, −5.02510728073351091840954352805, −4.22314860578040453376578568320, −3.66770761519845497396434538299, −3.29064182715467801872309384011, −2.10177746968632723446912062423, −0.52987945796703708076159945332,
0.52987945796703708076159945332, 2.10177746968632723446912062423, 3.29064182715467801872309384011, 3.66770761519845497396434538299, 4.22314860578040453376578568320, 5.02510728073351091840954352805, 5.73741301530147015786759785914, 6.76583759083629767409026960561, 7.41425636417035830373388925371, 8.146957276347753514201627006105
