| L(s) = 1 | − 4.40·2-s − 14.4·3-s − 12.5·4-s − 25·5-s + 63.4·6-s − 75.0·7-s + 196.·8-s − 35.5·9-s + 110.·10-s + 121·11-s + 181.·12-s + 337.·13-s + 330.·14-s + 360.·15-s − 463.·16-s + 2.17e3·17-s + 156.·18-s − 361·19-s + 314.·20-s + 1.08e3·21-s − 533.·22-s + 597.·23-s − 2.82e3·24-s + 625·25-s − 1.48e3·26-s + 4.01e3·27-s + 943.·28-s + ⋯ |
| L(s) = 1 | − 0.779·2-s − 0.923·3-s − 0.392·4-s − 0.447·5-s + 0.719·6-s − 0.578·7-s + 1.08·8-s − 0.146·9-s + 0.348·10-s + 0.301·11-s + 0.362·12-s + 0.553·13-s + 0.451·14-s + 0.413·15-s − 0.452·16-s + 1.82·17-s + 0.114·18-s − 0.229·19-s + 0.175·20-s + 0.534·21-s − 0.234·22-s + 0.235·23-s − 1.00·24-s + 0.200·25-s − 0.431·26-s + 1.05·27-s + 0.227·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8632864676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8632864676\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 4.40T + 32T^{2} \) |
| 3 | \( 1 + 14.4T + 243T^{2} \) |
| 7 | \( 1 + 75.0T + 1.68e4T^{2} \) |
| 13 | \( 1 - 337.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.17e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 597.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.97e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.58e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.55e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.17e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.40e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.62e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.71e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.74e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.24e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.11e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.28e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.84e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.37e4T + 8.58e9T^{2} \) |
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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
−9.237750967142653836316863510942, −8.360108591226448714352349321801, −7.72890867050147743678182756414, −6.64687851778914538190289598743, −5.86894886478953871789007611887, −4.94390955084878683418973133110, −3.99226361027269777591659187726, −2.91733942168919928917154549120, −1.06406965705017484624337635236, −0.63237741166012412365849278789,
0.63237741166012412365849278789, 1.06406965705017484624337635236, 2.91733942168919928917154549120, 3.99226361027269777591659187726, 4.94390955084878683418973133110, 5.86894886478953871789007611887, 6.64687851778914538190289598743, 7.72890867050147743678182756414, 8.360108591226448714352349321801, 9.237750967142653836316863510942
