| L(s) = 1 | + 2·2-s + 2.01·3-s + 4·4-s + 4.02·6-s + 29.3·7-s + 8·8-s − 22.9·9-s + 7.09·11-s + 8.04·12-s + 70.3·13-s + 58.6·14-s + 16·16-s − 32.6·17-s − 45.9·18-s + 153.·19-s + 59.0·21-s + 14.1·22-s − 143.·23-s + 16.0·24-s + 140.·26-s − 100.·27-s + 117.·28-s + 135.·29-s + 99.6·31-s + 32·32-s + 14.2·33-s − 65.2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.387·3-s + 0.5·4-s + 0.273·6-s + 1.58·7-s + 0.353·8-s − 0.850·9-s + 0.194·11-s + 0.193·12-s + 1.50·13-s + 1.11·14-s + 0.250·16-s − 0.465·17-s − 0.601·18-s + 1.85·19-s + 0.613·21-s + 0.137·22-s − 1.29·23-s + 0.136·24-s + 1.06·26-s − 0.716·27-s + 0.791·28-s + 0.864·29-s + 0.577·31-s + 0.176·32-s + 0.0753·33-s − 0.329·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.158123811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.158123811\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 - 2.01T + 27T^{2} \) |
| 7 | \( 1 - 29.3T + 343T^{2} \) |
| 11 | \( 1 - 7.09T + 1.33e3T^{2} \) |
| 13 | \( 1 - 70.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 32.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 153.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 143.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 99.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 336.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 162.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 85.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 367.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 349.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 130.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 334.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 473.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 306.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 377.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 320.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 509.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 472.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 85.4T + 9.12e5T^{2} \) |
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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
−9.105273492627077209233581177951, −8.320456429088299855397883729749, −7.902632559067613433561563416123, −6.76561880137902887652551076887, −5.73994711045035122655286420025, −5.14472417863319779373666243077, −4.08985953182208306743873491000, −3.26242246645149941900462572890, −2.08566130981062361501264725234, −1.11611725617890758826147615256,
1.11611725617890758826147615256, 2.08566130981062361501264725234, 3.26242246645149941900462572890, 4.08985953182208306743873491000, 5.14472417863319779373666243077, 5.73994711045035122655286420025, 6.76561880137902887652551076887, 7.902632559067613433561563416123, 8.320456429088299855397883729749, 9.105273492627077209233581177951
