| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 6.24·11-s + 12-s − 0.171·13-s + 16-s + 4.65·17-s + 18-s − 2.58·19-s − 6.24·22-s + 1.58·23-s + 24-s − 0.171·26-s + 27-s − 10.6·29-s − 7.24·31-s + 32-s − 6.24·33-s + 4.65·34-s + 36-s + 0.242·37-s − 2.58·38-s − 0.171·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.88·11-s + 0.288·12-s − 0.0475·13-s + 0.250·16-s + 1.12·17-s + 0.235·18-s − 0.593·19-s − 1.33·22-s + 0.330·23-s + 0.204·24-s − 0.0336·26-s + 0.192·27-s − 1.97·29-s − 1.30·31-s + 0.176·32-s − 1.08·33-s + 0.798·34-s + 0.166·36-s + 0.0398·37-s − 0.419·38-s − 0.0274·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 + 0.171T + 13T^{2} \) |
| 17 | \( 1 - 4.65T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 + 7.24T + 31T^{2} \) |
| 37 | \( 1 - 0.242T + 37T^{2} \) |
| 41 | \( 1 + 7T + 41T^{2} \) |
| 43 | \( 1 + 3.58T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 2.65T + 53T^{2} \) |
| 59 | \( 1 - 1.58T + 59T^{2} \) |
| 61 | \( 1 + 2.65T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 - 6.24T + 79T^{2} \) |
| 83 | \( 1 - 5.58T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−7.51730089191178291915861109009, −7.03716086821026916781904023683, −5.94275678577671511610126960829, −5.34094169723031133846583223340, −4.85777081866958334004534915661, −3.74382597420887414924443490557, −3.25609176130560697881516473228, −2.39871554531257663932178913878, −1.66293188553856289000286312559, 0,
1.66293188553856289000286312559, 2.39871554531257663932178913878, 3.25609176130560697881516473228, 3.74382597420887414924443490557, 4.85777081866958334004534915661, 5.34094169723031133846583223340, 5.94275678577671511610126960829, 7.03716086821026916781904023683, 7.51730089191178291915861109009
