| L(s) = 1 | − 2·2-s + 5.31·3-s + 4·4-s − 10.6·6-s − 34.1·7-s − 8·8-s + 1.23·9-s − 58.9·11-s + 21.2·12-s − 57.0·13-s + 68.3·14-s + 16·16-s + 116.·17-s − 2.46·18-s − 35.0·19-s − 181.·21-s + 117.·22-s − 23·23-s − 42.5·24-s + 114.·26-s − 136.·27-s − 136.·28-s − 171.·29-s + 131.·31-s − 32·32-s − 313.·33-s − 233.·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.02·3-s + 0.5·4-s − 0.723·6-s − 1.84·7-s − 0.353·8-s + 0.0456·9-s − 1.61·11-s + 0.511·12-s − 1.21·13-s + 1.30·14-s + 0.250·16-s + 1.66·17-s − 0.0322·18-s − 0.422·19-s − 1.88·21-s + 1.14·22-s − 0.208·23-s − 0.361·24-s + 0.860·26-s − 0.975·27-s − 0.922·28-s − 1.09·29-s + 0.760·31-s − 0.176·32-s − 1.65·33-s − 1.17·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8350215329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8350215329\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 5.31T + 27T^{2} \) |
| 7 | \( 1 + 34.1T + 343T^{2} \) |
| 11 | \( 1 + 58.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 57.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 35.0T + 6.85e3T^{2} \) |
| 29 | \( 1 + 171.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 131.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 325.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 72.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 59.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 479.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 230.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 495.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 267.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 11.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 353.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 466.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 773.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 22.4T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.17e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.31e3T + 9.12e5T^{2} \) |
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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
−9.623714167174792851177424570843, −8.659887763500979759157391677402, −7.69993874230195372860499336786, −7.41422580961081607971547532714, −6.15334361837555262850015111752, −5.37626767159165499267258448769, −3.73394930860683520945183223871, −2.81512173472425835089793273199, −2.44449001743607572378315952792, −0.45964836575715670988544389568,
0.45964836575715670988544389568, 2.44449001743607572378315952792, 2.81512173472425835089793273199, 3.73394930860683520945183223871, 5.37626767159165499267258448769, 6.15334361837555262850015111752, 7.41422580961081607971547532714, 7.69993874230195372860499336786, 8.659887763500979759157391677402, 9.623714167174792851177424570843
