| L(s) = 1 | − 2.14·2-s − 3.39·4-s + 23.2·7-s + 24.4·8-s − 12.4·11-s + 50.6·13-s − 49.9·14-s − 25.3·16-s + 63.6·17-s + 5.41·19-s + 26.8·22-s − 88.4·23-s − 108.·26-s − 78.9·28-s − 144.·29-s − 277.·31-s − 141.·32-s − 136.·34-s − 243.·37-s − 11.6·38-s − 468.·41-s + 498.·43-s + 42.4·44-s + 189.·46-s + 452.·47-s + 197.·49-s − 171.·52-s + ⋯ |
| L(s) = 1 | − 0.758·2-s − 0.424·4-s + 1.25·7-s + 1.08·8-s − 0.342·11-s + 1.07·13-s − 0.952·14-s − 0.395·16-s + 0.908·17-s + 0.0653·19-s + 0.259·22-s − 0.802·23-s − 0.819·26-s − 0.532·28-s − 0.925·29-s − 1.61·31-s − 0.780·32-s − 0.689·34-s − 1.08·37-s − 0.0495·38-s − 1.78·41-s + 1.76·43-s + 0.145·44-s + 0.608·46-s + 1.40·47-s + 0.576·49-s − 0.458·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.14T + 8T^{2} \) |
| 7 | \( 1 - 23.2T + 343T^{2} \) |
| 11 | \( 1 + 12.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 50.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 5.41T + 6.85e3T^{2} \) |
| 23 | \( 1 + 88.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 144.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 277.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 243.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 468.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 498.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 452.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 223.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 115.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 82.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 810.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 134.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 707.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 417.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 922.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 114.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 361.T + 9.12e5T^{2} \) |
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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
−8.525117228597476766419913666009, −7.71566949850348026806689856168, −7.30259214474057038808240437256, −5.82005722095893237037580080233, −5.27087258286984514936902775380, −4.28251700714461722391615712513, −3.51002173076041975493788336490, −1.89285836251520250132303775294, −1.26074859553949280278086431834, 0,
1.26074859553949280278086431834, 1.89285836251520250132303775294, 3.51002173076041975493788336490, 4.28251700714461722391615712513, 5.27087258286984514936902775380, 5.82005722095893237037580080233, 7.30259214474057038808240437256, 7.71566949850348026806689856168, 8.525117228597476766419913666009
