| L(s) = 1 | − 3.21·2-s + 2.35·4-s − 7·7-s + 18.1·8-s + 2.09·11-s + 80.8·13-s + 22.5·14-s − 77.2·16-s − 101.·17-s + 143.·19-s − 6.75·22-s + 116.·23-s − 260.·26-s − 16.4·28-s − 181.·29-s + 303.·31-s + 103.·32-s + 328.·34-s + 158.·37-s − 461.·38-s + 379.·41-s − 238.·43-s + 4.93·44-s − 374.·46-s − 125.·47-s + 49·49-s + 190.·52-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 0.293·4-s − 0.377·7-s + 0.803·8-s + 0.0575·11-s + 1.72·13-s + 0.429·14-s − 1.20·16-s − 1.45·17-s + 1.73·19-s − 0.0654·22-s + 1.05·23-s − 1.96·26-s − 0.111·28-s − 1.16·29-s + 1.75·31-s + 0.570·32-s + 1.65·34-s + 0.703·37-s − 1.96·38-s + 1.44·41-s − 0.847·43-s + 0.0169·44-s − 1.19·46-s − 0.390·47-s + 0.142·49-s + 0.506·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.139325788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.139325788\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 3.21T + 8T^{2} \) |
| 11 | \( 1 - 2.09T + 1.33e3T^{2} \) |
| 13 | \( 1 - 80.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 181.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 303.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 158.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 379.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 238.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 125.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 43.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 31.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 812.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 426.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 471.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.20e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 886.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 134.T + 9.12e5T^{2} \) |
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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
−9.207657532742788896673708399949, −8.404303254395056572703653584064, −7.68018920203486724185820607451, −6.80538481968834873165096169717, −6.05292600199169286630496888820, −4.88856378236594051449129067490, −3.94197594048038654966427972767, −2.86004262593839460654395798177, −1.47238976308111370758881928731, −0.67810971577713666097825071591,
0.67810971577713666097825071591, 1.47238976308111370758881928731, 2.86004262593839460654395798177, 3.94197594048038654966427972767, 4.88856378236594051449129067490, 6.05292600199169286630496888820, 6.80538481968834873165096169717, 7.68018920203486724185820607451, 8.404303254395056572703653584064, 9.207657532742788896673708399949
