| L(s) = 1 | + 2.65·2-s + 5.05·4-s − 5-s + 8.10·8-s − 2.65·10-s + 4.39·11-s − 3.39·13-s + 11.4·16-s + 2.65·17-s + 7.36·19-s − 5.05·20-s + 11.6·22-s − 4.75·23-s + 25-s − 9.01·26-s − 3.70·29-s − 1.51·31-s + 14.1·32-s + 7.05·34-s − 0.447·37-s + 19.5·38-s − 8.10·40-s + 6.49·41-s − 1.13·43-s + 22.2·44-s − 12.6·46-s + 2.51·47-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 2.52·4-s − 0.447·5-s + 2.86·8-s − 0.839·10-s + 1.32·11-s − 0.941·13-s + 2.85·16-s + 0.644·17-s + 1.68·19-s − 1.12·20-s + 2.48·22-s − 0.992·23-s + 0.200·25-s − 1.76·26-s − 0.688·29-s − 0.272·31-s + 2.49·32-s + 1.20·34-s − 0.0735·37-s + 3.17·38-s − 1.28·40-s + 1.01·41-s − 0.173·43-s + 3.34·44-s − 1.86·46-s + 0.367·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.963088386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.963088386\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.65T + 2T^{2} \) |
| 11 | \( 1 - 4.39T + 11T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 17 | \( 1 - 2.65T + 17T^{2} \) |
| 19 | \( 1 - 7.36T + 19T^{2} \) |
| 23 | \( 1 + 4.75T + 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 + 1.51T + 31T^{2} \) |
| 37 | \( 1 + 0.447T + 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 + 1.13T + 43T^{2} \) |
| 47 | \( 1 - 2.51T + 47T^{2} \) |
| 53 | \( 1 - 5.31T + 53T^{2} \) |
| 59 | \( 1 + 14.6T + 59T^{2} \) |
| 61 | \( 1 - 8.25T + 61T^{2} \) |
| 67 | \( 1 + 7.91T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 2.57T + 79T^{2} \) |
| 83 | \( 1 + 0.552T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 2.53T + 97T^{2} \) |
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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
−9.166770769754420441078649853353, −7.70987950923819347837802340606, −7.38207965934616898965362116630, −6.48414173745127614906176002616, −5.70206618416265998495502169801, −5.01793186626411882905544316449, −4.09237581755165717719229585492, −3.56217528895203118908991601570, −2.62845001119357909321333148285, −1.41217717637559608976803786428,
1.41217717637559608976803786428, 2.62845001119357909321333148285, 3.56217528895203118908991601570, 4.09237581755165717719229585492, 5.01793186626411882905544316449, 5.70206618416265998495502169801, 6.48414173745127614906176002616, 7.38207965934616898965362116630, 7.70987950923819347837802340606, 9.166770769754420441078649853353
