| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 7.80·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s + 15.6·10-s + 22.4·11-s − 12·12-s − 72.9·13-s + 14·14-s − 23.4·15-s + 16·16-s − 126.·17-s + 18·18-s − 116.·19-s + 31.2·20-s − 21·21-s + 44.9·22-s + 23·23-s − 24·24-s − 64.0·25-s − 145.·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.698·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.493·10-s + 0.616·11-s − 0.288·12-s − 1.55·13-s + 0.267·14-s − 0.403·15-s + 0.250·16-s − 1.80·17-s + 0.235·18-s − 1.40·19-s + 0.349·20-s − 0.218·21-s + 0.435·22-s + 0.208·23-s − 0.204·24-s − 0.512·25-s − 1.10·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 7.80T + 125T^{2} \) |
| 11 | \( 1 - 22.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 72.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 126.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 131.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 39.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 84.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 233.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 20.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 56.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 417.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 250.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 248.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 521.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 398.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 312.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 61.2T + 4.93e5T^{2} \) |
| 83 | \( 1 + 135.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 66.1T + 7.04e5T^{2} \) |
| 97 | \( 1 - 904.T + 9.12e5T^{2} \) |
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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
−9.321056008473067318543577761977, −8.410497661156933922711239668748, −7.08626802656413908228516877325, −6.60573326637664732682723228717, −5.69145814751685375927291699600, −4.74669331733522724934132497005, −4.19426045344816364248279350979, −2.52586782288972588750359632974, −1.77317120180234963311821162415, 0,
1.77317120180234963311821162415, 2.52586782288972588750359632974, 4.19426045344816364248279350979, 4.74669331733522724934132497005, 5.69145814751685375927291699600, 6.60573326637664732682723228717, 7.08626802656413908228516877325, 8.410497661156933922711239668748, 9.321056008473067318543577761977
