| L(s) = 1 | − 1.77·2-s + 0.241·3-s + 2.13·4-s − 5-s − 0.426·6-s + 0.709·7-s − 2.01·8-s − 0.941·9-s + 1.77·10-s + 1.49·11-s + 0.514·12-s + 13-s − 1.25·14-s − 0.241·15-s + 1.42·16-s + 1.13·17-s + 1.66·18-s − 2.13·20-s + 0.170·21-s − 2.65·22-s − 1.94·23-s − 0.485·24-s + 25-s − 1.77·26-s − 0.468·27-s + 1.51·28-s + 0.426·30-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 0.241·3-s + 2.13·4-s − 5-s − 0.426·6-s + 0.709·7-s − 2.01·8-s − 0.941·9-s + 1.77·10-s + 1.49·11-s + 0.514·12-s + 13-s − 1.25·14-s − 0.241·15-s + 1.42·16-s + 1.13·17-s + 1.66·18-s − 2.13·20-s + 0.170·21-s − 2.65·22-s − 1.94·23-s − 0.485·24-s + 25-s − 1.77·26-s − 0.468·27-s + 1.51·28-s + 0.426·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5280540241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5280540241\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.77T + T^{2} \) |
| 3 | \( 1 - 0.241T + T^{2} \) |
| 7 | \( 1 - 0.709T + T^{2} \) |
| 11 | \( 1 - 1.49T + T^{2} \) |
| 17 | \( 1 - 1.13T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.94T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.77T + T^{2} \) |
| 47 | \( 1 - 1.94T + T^{2} \) |
| 53 | \( 1 + 1.49T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.49T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.709T + T^{2} \) |
| 97 | \( 1 + 0.241T + T^{2} \) |
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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
−9.051184085132912562182619074287, −8.639920265666697508932896082356, −7.88644843308622466108949761332, −7.56484800610507271181940937859, −6.43793585164427383525443421257, −5.74428313387025972180445443878, −4.12437415305407235023098340586, −3.38615276817347851932622009752, −2.01588216875098740246716907165, −0.960778428313051497684955063666,
0.960778428313051497684955063666, 2.01588216875098740246716907165, 3.38615276817347851932622009752, 4.12437415305407235023098340586, 5.74428313387025972180445443878, 6.43793585164427383525443421257, 7.56484800610507271181940937859, 7.88644843308622466108949761332, 8.639920265666697508932896082356, 9.051184085132912562182619074287
