| L(s) = 1 | + 2-s − 3.03·3-s + 4-s + 2.03·5-s − 3.03·6-s − 7-s + 8-s + 6.23·9-s + 2.03·10-s − 0.723·11-s − 3.03·12-s − 6.23·13-s − 14-s − 6.19·15-s + 16-s + 5.76·17-s + 6.23·18-s − 2.31·19-s + 2.03·20-s + 3.03·21-s − 0.723·22-s − 3.03·24-s − 0.840·25-s − 6.23·26-s − 9.84·27-s − 28-s + 6.76·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.75·3-s + 0.5·4-s + 0.912·5-s − 1.24·6-s − 0.377·7-s + 0.353·8-s + 2.07·9-s + 0.644·10-s − 0.218·11-s − 0.877·12-s − 1.73·13-s − 0.267·14-s − 1.60·15-s + 0.250·16-s + 1.39·17-s + 1.47·18-s − 0.531·19-s + 0.456·20-s + 0.663·21-s − 0.154·22-s − 0.620·24-s − 0.168·25-s − 1.22·26-s − 1.89·27-s − 0.188·28-s + 1.25·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 3.03T + 3T^{2} \) |
| 5 | \( 1 - 2.03T + 5T^{2} \) |
| 11 | \( 1 + 0.723T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 - 5.76T + 17T^{2} \) |
| 19 | \( 1 + 2.31T + 19T^{2} \) |
| 29 | \( 1 - 6.76T + 29T^{2} \) |
| 31 | \( 1 + 8.27T + 31T^{2} \) |
| 37 | \( 1 - 6.27T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4.39T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 + 8.76T + 53T^{2} \) |
| 59 | \( 1 + 4.51T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 5.96T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 5.63T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 1.23T + 83T^{2} \) |
| 89 | \( 1 + 1.11T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.39732725868166362263014283528, −6.51032169401891029769954010822, −5.89546290397040831935135083928, −5.65411540849899920634489498322, −4.78991377693115944107158170809, −4.42174429948420238833500697640, −3.13641913677285138658293160700, −2.24354529104947159801449193325, −1.22608784023212553541723377258, 0,
1.22608784023212553541723377258, 2.24354529104947159801449193325, 3.13641913677285138658293160700, 4.42174429948420238833500697640, 4.78991377693115944107158170809, 5.65411540849899920634489498322, 5.89546290397040831935135083928, 6.51032169401891029769954010822, 7.39732725868166362263014283528
