| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s − 2·9-s + 10-s + 12-s − 5·13-s + 14-s − 15-s + 16-s − 6·17-s + 2·18-s − 5·19-s − 20-s − 21-s − 3·23-s − 24-s + 25-s + 5·26-s − 5·27-s − 28-s + 30-s − 10·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.288·12-s − 1.38·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.471·18-s − 1.14·19-s − 0.223·20-s − 0.218·21-s − 0.625·23-s − 0.204·24-s + 1/5·25-s + 0.980·26-s − 0.962·27-s − 0.188·28-s + 0.182·30-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3167985133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3167985133\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.979271532207034515596997684652, −7.09253926729061395001301455703, −6.76251320478139418479371066078, −5.83048622129759522333281730379, −5.00363780163613374507889571120, −4.08816586770610846447665335708, −3.38566792649676562885736808929, −2.33333236574734846582713446503, −2.09927748430030322621799266461, −0.27062761318745754080750884922,
0.27062761318745754080750884922, 2.09927748430030322621799266461, 2.33333236574734846582713446503, 3.38566792649676562885736808929, 4.08816586770610846447665335708, 5.00363780163613374507889571120, 5.83048622129759522333281730379, 6.76251320478139418479371066078, 7.09253926729061395001301455703, 7.979271532207034515596997684652
