| L(s) = 1 | − 1.77·2-s + 3·3-s − 4.84·4-s + 6.23·5-s − 5.32·6-s − 18.0·7-s + 22.8·8-s + 9·9-s − 11.0·10-s + 13.3·11-s − 14.5·12-s − 66.3·13-s + 32.0·14-s + 18.6·15-s − 1.68·16-s − 97.6·17-s − 15.9·18-s + 109.·19-s − 30.2·20-s − 54.2·21-s − 23.6·22-s − 147.·23-s + 68.4·24-s − 86.1·25-s + 117.·26-s + 27·27-s + 87.6·28-s + ⋯ |
| L(s) = 1 | − 0.627·2-s + 0.577·3-s − 0.606·4-s + 0.557·5-s − 0.362·6-s − 0.976·7-s + 1.00·8-s + 0.333·9-s − 0.349·10-s + 0.365·11-s − 0.349·12-s − 1.41·13-s + 0.612·14-s + 0.321·15-s − 0.0263·16-s − 1.39·17-s − 0.209·18-s + 1.32·19-s − 0.337·20-s − 0.563·21-s − 0.229·22-s − 1.33·23-s + 0.581·24-s − 0.689·25-s + 0.888·26-s + 0.192·27-s + 0.591·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 59 | \( 1 + 59T \) |
| good | 2 | \( 1 + 1.77T + 8T^{2} \) |
| 5 | \( 1 - 6.23T + 125T^{2} \) |
| 7 | \( 1 + 18.0T + 343T^{2} \) |
| 11 | \( 1 - 13.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 66.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 97.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 109.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 147.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 173.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 148.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 446.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 182.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 223.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 529.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 398.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 788.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 288.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 139.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 549.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 190.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 410.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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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
−11.78418271443563426370521481313, −10.07829152469800530248039957614, −9.734208686093020312042537359052, −8.942362279301094827046087176026, −7.75265052060360502413297237569, −6.68340614886887603538688128090, −5.12993932603101210106068055557, −3.72752084068713117923020223843, −2.06797750947095367109343158962, 0,
2.06797750947095367109343158962, 3.72752084068713117923020223843, 5.12993932603101210106068055557, 6.68340614886887603538688128090, 7.75265052060360502413297237569, 8.942362279301094827046087176026, 9.734208686093020312042537359052, 10.07829152469800530248039957614, 11.78418271443563426370521481313
