| L(s) = 1 | + 1.49·2-s − 1.13·3-s + 1.24·4-s − 5-s − 1.70·6-s + 1.94·7-s + 0.360·8-s + 0.290·9-s − 1.49·10-s + 1.77·11-s − 1.41·12-s − 13-s + 2.90·14-s + 1.13·15-s − 0.700·16-s − 0.241·17-s + 0.435·18-s − 1.24·20-s − 2.20·21-s + 2.65·22-s + 0.709·23-s − 0.410·24-s + 25-s − 1.49·26-s + 0.805·27-s + 2.41·28-s + 1.70·30-s + ⋯ |
| L(s) = 1 | + 1.49·2-s − 1.13·3-s + 1.24·4-s − 5-s − 1.70·6-s + 1.94·7-s + 0.360·8-s + 0.290·9-s − 1.49·10-s + 1.77·11-s − 1.41·12-s − 13-s + 2.90·14-s + 1.13·15-s − 0.700·16-s − 0.241·17-s + 0.435·18-s − 1.24·20-s − 2.20·21-s + 2.65·22-s + 0.709·23-s − 0.410·24-s + 25-s − 1.49·26-s + 0.805·27-s + 2.41·28-s + 1.70·30-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.848791501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.848791501\) |
| \(L(1)\) |
|
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 | \( 1 - 1.49T + T^{2} \) |
| 3 | \( 1 + 1.13T + T^{2} \) |
| 7 | \( 1 - 1.94T + T^{2} \) |
| 11 | \( 1 - 1.77T + T^{2} \) |
| 17 | \( 1 + 0.241T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 0.709T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.49T + T^{2} \) |
| 47 | \( 1 - 0.709T + T^{2} \) |
| 53 | \( 1 + 1.77T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.77T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.94T + T^{2} \) |
| 97 | \( 1 + 1.13T + T^{2} \) |
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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
−9.228166763453104265125171283880, −8.433593131186528001335535505861, −7.42363926918964692259661812601, −6.78782745693208986438556680079, −5.91312043643718349011133094630, −5.06889280235961583805290607444, −4.50394950790259513652408786338, −4.13240535960513333598165367327, −2.76464103271535897070692842725, −1.28824742785316489609918592529,
1.28824742785316489609918592529, 2.76464103271535897070692842725, 4.13240535960513333598165367327, 4.50394950790259513652408786338, 5.06889280235961583805290607444, 5.91312043643718349011133094630, 6.78782745693208986438556680079, 7.42363926918964692259661812601, 8.433593131186528001335535505861, 9.228166763453104265125171283880
