| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 14.0·5-s + 6·6-s + 1.83·7-s + 8·8-s + 9·9-s − 28.0·10-s − 65.5·11-s + 12·12-s + 80.7·13-s + 3.66·14-s − 42.0·15-s + 16·16-s + 17.9·17-s + 18·18-s − 56.0·20-s + 5.49·21-s − 131.·22-s + 107.·23-s + 24·24-s + 71.3·25-s + 161.·26-s + 27·27-s + 7.32·28-s − 183.·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.25·5-s + 0.408·6-s + 0.0988·7-s + 0.353·8-s + 0.333·9-s − 0.886·10-s − 1.79·11-s + 0.288·12-s + 1.72·13-s + 0.0698·14-s − 0.723·15-s + 0.250·16-s + 0.256·17-s + 0.235·18-s − 0.626·20-s + 0.0570·21-s − 1.27·22-s + 0.970·23-s + 0.204·24-s + 0.571·25-s + 1.21·26-s + 0.192·27-s + 0.0494·28-s − 1.17·29-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 | \( 1 + 14.0T + 125T^{2} \) |
| 7 | \( 1 - 1.83T + 343T^{2} \) |
| 11 | \( 1 + 65.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 80.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.9T + 4.91e3T^{2} \) |
| 23 | \( 1 - 107.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 183.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 105.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 294.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 325.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 88.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 416.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 292.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 43.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 711.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 953.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 753.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 201.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 865.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 642.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.42e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 264.T + 9.12e5T^{2} \) |
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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
−8.289638694855021178265011267953, −7.56613716924874031391739772603, −6.95903915461875754492819292569, −5.77317791069472344334960403579, −5.05174173145529113148923315510, −4.06787316897770479769281803262, −3.43401628463443164060872049770, −2.71712710818037177193966661228, −1.39752703304018673778161682172, 0,
1.39752703304018673778161682172, 2.71712710818037177193966661228, 3.43401628463443164060872049770, 4.06787316897770479769281803262, 5.05174173145529113148923315510, 5.77317791069472344334960403579, 6.95903915461875754492819292569, 7.56613716924874031391739772603, 8.289638694855021178265011267953
