| L(s) = 1 | + 2·2-s − 1.85·3-s + 4·4-s + 2.80·5-s − 3.70·6-s + 8·8-s − 23.5·9-s + 5.61·10-s + 18.6·11-s − 7.41·12-s + 39.9·13-s − 5.20·15-s + 16·16-s − 34.3·17-s − 47.1·18-s − 60.1·19-s + 11.2·20-s + 37.2·22-s + 23·23-s − 14.8·24-s − 117.·25-s + 79.8·26-s + 93.7·27-s + 145.·29-s − 10.4·30-s + 94.1·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.356·3-s + 0.5·4-s + 0.251·5-s − 0.252·6-s + 0.353·8-s − 0.872·9-s + 0.177·10-s + 0.510·11-s − 0.178·12-s + 0.851·13-s − 0.0896·15-s + 0.250·16-s − 0.490·17-s − 0.617·18-s − 0.726·19-s + 0.125·20-s + 0.360·22-s + 0.208·23-s − 0.126·24-s − 0.936·25-s + 0.602·26-s + 0.667·27-s + 0.929·29-s − 0.0633·30-s + 0.545·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.118309627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.118309627\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 1.85T + 27T^{2} \) |
| 5 | \( 1 - 2.80T + 125T^{2} \) |
| 11 | \( 1 - 18.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 39.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 34.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 60.1T + 6.85e3T^{2} \) |
| 29 | \( 1 - 145.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 94.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 43.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 206.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 153.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 274.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 609.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 495.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 818.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 616.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 975.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 149.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 272.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.18T + 5.71e5T^{2} \) |
| 89 | \( 1 - 150.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.66e3T + 9.12e5T^{2} \) |
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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
−8.586689439843412674585189739827, −7.965342821494651768256350216393, −6.71046542070324724752309980549, −6.27651010452455007833356329236, −5.61233191857496076655544097904, −4.67797844902374162534086551059, −3.88561065478443452598560419267, −2.91704602011248500386106659046, −1.95574870030225340385917761076, −0.71996486831932671289526236948,
0.71996486831932671289526236948, 1.95574870030225340385917761076, 2.91704602011248500386106659046, 3.88561065478443452598560419267, 4.67797844902374162534086551059, 5.61233191857496076655544097904, 6.27651010452455007833356329236, 6.71046542070324724752309980549, 7.965342821494651768256350216393, 8.586689439843412674585189739827
