| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 14.2·5-s − 6·6-s + 13.2·7-s + 8·8-s + 9·9-s + 28.5·10-s − 65.9·11-s − 12·12-s − 68.5·13-s + 26.5·14-s − 42.7·15-s + 16·16-s + 99.6·17-s + 18·18-s + 57.0·20-s − 39.7·21-s − 131.·22-s + 3.53·23-s − 24·24-s + 78.1·25-s − 137.·26-s − 27·27-s + 53.0·28-s − 81.9·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.27·5-s − 0.408·6-s + 0.715·7-s + 0.353·8-s + 0.333·9-s + 0.901·10-s − 1.80·11-s − 0.288·12-s − 1.46·13-s + 0.506·14-s − 0.736·15-s + 0.250·16-s + 1.42·17-s + 0.235·18-s + 0.637·20-s − 0.413·21-s − 1.27·22-s + 0.0320·23-s − 0.204·24-s + 0.625·25-s − 1.03·26-s − 0.192·27-s + 0.357·28-s − 0.524·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 14.2T + 125T^{2} \) |
| 7 | \( 1 - 13.2T + 343T^{2} \) |
| 11 | \( 1 + 65.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 68.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 99.6T + 4.91e3T^{2} \) |
| 23 | \( 1 - 3.53T + 1.21e4T^{2} \) |
| 29 | \( 1 + 81.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 247.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 421.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 345.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 366.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 90.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 688.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 182.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 0.517T + 2.26e5T^{2} \) |
| 67 | \( 1 + 159.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 791.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 322.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 318.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 684.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 720.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 415.T + 9.12e5T^{2} \) |
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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
−7.87839943554406021193752701519, −7.65817198439213465915424773738, −6.54762569082630123050346536011, −5.58988454803544394526593744880, −5.24178893722597618389588252241, −4.74116751891594191722940124403, −3.22213868692223366611329003916, −2.31377187247067848000562867497, −1.56419415318510693927094167028, 0,
1.56419415318510693927094167028, 2.31377187247067848000562867497, 3.22213868692223366611329003916, 4.74116751891594191722940124403, 5.24178893722597618389588252241, 5.58988454803544394526593744880, 6.54762569082630123050346536011, 7.65817198439213465915424773738, 7.87839943554406021193752701519
