| L(s) = 1 | + 4.70·2-s + 3·3-s + 14.1·4-s + 14.1·6-s − 16.2·7-s + 28.7·8-s + 9·9-s − 40.2·11-s + 42.3·12-s + 19.7·13-s − 76.2·14-s + 22.1·16-s + 83.0·17-s + 42.3·18-s − 48.8·19-s − 48.6·21-s − 189.·22-s + 1.61·23-s + 86.1·24-s + 93.0·26-s + 27·27-s − 228.·28-s − 24.5·29-s − 12.4·31-s − 125.·32-s − 120.·33-s + 390.·34-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 0.577·3-s + 1.76·4-s + 0.959·6-s − 0.875·7-s + 1.26·8-s + 0.333·9-s − 1.10·11-s + 1.01·12-s + 0.422·13-s − 1.45·14-s + 0.345·16-s + 1.18·17-s + 0.554·18-s − 0.589·19-s − 0.505·21-s − 1.83·22-s + 0.0146·23-s + 0.732·24-s + 0.701·26-s + 0.192·27-s − 1.54·28-s − 0.157·29-s − 0.0719·31-s − 0.694·32-s − 0.636·33-s + 1.96·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.687263950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.687263950\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 4.70T + 8T^{2} \) |
| 7 | \( 1 + 16.2T + 343T^{2} \) |
| 11 | \( 1 + 40.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 83.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 48.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 1.61T + 1.21e4T^{2} \) |
| 29 | \( 1 + 24.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 12.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 325.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 242.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 367.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 204.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 61.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 112.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 477.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 558.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 558.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 96.9T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3T + 9.12e5T^{2} \) |
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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
−13.90911486129079565122183840125, −13.02869188257218953584293039039, −12.46361857535817110721507238677, −11.00006750586232230937605439277, −9.705511164404952700857363636706, −7.964995495075949724713933550451, −6.54672827298846704063399958730, −5.34165747804085571998374927723, −3.79684465948635158134660234263, −2.67789297042972100403261059468,
2.67789297042972100403261059468, 3.79684465948635158134660234263, 5.34165747804085571998374927723, 6.54672827298846704063399958730, 7.964995495075949724713933550451, 9.705511164404952700857363636706, 11.00006750586232230937605439277, 12.46361857535817110721507238677, 13.02869188257218953584293039039, 13.90911486129079565122183840125
