| L(s) = 1 | + 2-s − 3-s + 4-s − 0.825·5-s − 6-s + 7-s + 8-s + 9-s − 0.825·10-s − 0.576·11-s − 12-s − 13-s + 14-s + 0.825·15-s + 16-s − 17-s + 18-s + 1.54·19-s − 0.825·20-s − 21-s − 0.576·22-s − 6.61·23-s − 24-s − 4.31·25-s − 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.369·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.260·10-s − 0.173·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.213·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.354·19-s − 0.184·20-s − 0.218·21-s − 0.122·22-s − 1.37·23-s − 0.204·24-s − 0.863·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 0.825T + 5T^{2} \) |
| 11 | \( 1 + 0.576T + 11T^{2} \) |
| 19 | \( 1 - 1.54T + 19T^{2} \) |
| 23 | \( 1 + 6.61T + 23T^{2} \) |
| 29 | \( 1 - 5.19T + 29T^{2} \) |
| 31 | \( 1 - 6.69T + 31T^{2} \) |
| 37 | \( 1 - 5.19T + 37T^{2} \) |
| 41 | \( 1 + 6.84T + 41T^{2} \) |
| 43 | \( 1 - 4.37T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 4.72T + 53T^{2} \) |
| 59 | \( 1 + 7.77T + 59T^{2} \) |
| 61 | \( 1 - 0.248T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 4.34T + 71T^{2} \) |
| 73 | \( 1 + 2.22T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 - 2.84T + 89T^{2} \) |
| 97 | \( 1 + 6.59T + 97T^{2} \) |
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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
−7.26218011702510112487913265913, −6.51038229353129714385561432995, −6.00396911669930422532204022209, −5.22537482862346745183346842394, −4.57583217399393973944463792835, −4.07261591290752982768957262612, −3.13716400003829042634456405802, −2.26349861294258869908660718236, −1.30760759718658916873677594916, 0,
1.30760759718658916873677594916, 2.26349861294258869908660718236, 3.13716400003829042634456405802, 4.07261591290752982768957262612, 4.57583217399393973944463792835, 5.22537482862346745183346842394, 6.00396911669930422532204022209, 6.51038229353129714385561432995, 7.26218011702510112487913265913
