| L(s) = 1 | − 2.61·2-s − 3-s + 4.85·4-s − 5-s + 2.61·6-s + 1.23·7-s − 7.47·8-s − 2·9-s + 2.61·10-s − 3.23·11-s − 4.85·12-s − 6.23·13-s − 3.23·14-s + 15-s + 9.85·16-s + 2.47·17-s + 5.23·18-s − 5.70·19-s − 4.85·20-s − 1.23·21-s + 8.47·22-s − 23-s + 7.47·24-s + 25-s + 16.3·26-s + 5·27-s + 6.00·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 0.577·3-s + 2.42·4-s − 0.447·5-s + 1.06·6-s + 0.467·7-s − 2.64·8-s − 0.666·9-s + 0.827·10-s − 0.975·11-s − 1.40·12-s − 1.72·13-s − 0.864·14-s + 0.258·15-s + 2.46·16-s + 0.599·17-s + 1.23·18-s − 1.30·19-s − 1.08·20-s − 0.269·21-s + 1.80·22-s − 0.208·23-s + 1.52·24-s + 0.200·25-s + 3.20·26-s + 0.962·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 29 | \( 1 + 0.527T + 29T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 + 9.70T + 37T^{2} \) |
| 41 | \( 1 + 7.47T + 41T^{2} \) |
| 43 | \( 1 - 3.70T + 43T^{2} \) |
| 47 | \( 1 - 9.47T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 + 6.23T + 71T^{2} \) |
| 73 | \( 1 + 6.70T + 73T^{2} \) |
| 79 | \( 1 - 8.76T + 79T^{2} \) |
| 83 | \( 1 + 6.47T + 83T^{2} \) |
| 89 | \( 1 - 2.76T + 89T^{2} \) |
| 97 | \( 1 + 6.18T + 97T^{2} \) |
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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
−12.41299076453725953754550936621, −11.68470866415916587033008191573, −10.68569453507544161612059950237, −10.01118667745436500401957574714, −8.603493233501857420483830647330, −7.86136208353759043663879641278, −6.80983195663352852256465199092, −5.24824667529767515960464977819, −2.46890853303187974438036531558, 0,
2.46890853303187974438036531558, 5.24824667529767515960464977819, 6.80983195663352852256465199092, 7.86136208353759043663879641278, 8.603493233501857420483830647330, 10.01118667745436500401957574714, 10.68569453507544161612059950237, 11.68470866415916587033008191573, 12.41299076453725953754550936621
