| L(s) = 1 | − 2.13·2-s − 2.23·3-s + 2.54·4-s + 4.75·6-s − 1.48·7-s − 1.16·8-s + 1.97·9-s + 4.68·11-s − 5.68·12-s + 0.361·13-s + 3.16·14-s − 2.60·16-s + 5.47·17-s − 4.21·18-s + 3.31·21-s − 9.98·22-s + 6.16·23-s + 2.60·24-s − 0.769·26-s + 2.28·27-s − 3.78·28-s − 0.895·29-s + 4.80·31-s + 7.89·32-s − 10.4·33-s − 11.6·34-s + 5.02·36-s + ⋯ |
| L(s) = 1 | − 1.50·2-s − 1.28·3-s + 1.27·4-s + 1.94·6-s − 0.561·7-s − 0.412·8-s + 0.658·9-s + 1.41·11-s − 1.63·12-s + 0.100·13-s + 0.846·14-s − 0.651·16-s + 1.32·17-s − 0.992·18-s + 0.723·21-s − 2.12·22-s + 1.28·23-s + 0.531·24-s − 0.150·26-s + 0.440·27-s − 0.715·28-s − 0.166·29-s + 0.862·31-s + 1.39·32-s − 1.81·33-s − 2.00·34-s + 0.838·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2.13T + 2T^{2} \) |
| 3 | \( 1 + 2.23T + 3T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 11 | \( 1 - 4.68T + 11T^{2} \) |
| 13 | \( 1 - 0.361T + 13T^{2} \) |
| 17 | \( 1 - 5.47T + 17T^{2} \) |
| 23 | \( 1 - 6.16T + 23T^{2} \) |
| 29 | \( 1 + 0.895T + 29T^{2} \) |
| 31 | \( 1 - 4.80T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 6.52T + 53T^{2} \) |
| 59 | \( 1 + 9.80T + 59T^{2} \) |
| 61 | \( 1 - 2.34T + 61T^{2} \) |
| 67 | \( 1 + 8.85T + 67T^{2} \) |
| 71 | \( 1 - 6.41T + 71T^{2} \) |
| 73 | \( 1 + 2.86T + 73T^{2} \) |
| 79 | \( 1 - 2.06T + 79T^{2} \) |
| 83 | \( 1 + 6.16T + 83T^{2} \) |
| 89 | \( 1 + 3.32T + 89T^{2} \) |
| 97 | \( 1 + 4.71T + 97T^{2} \) |
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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
−7.25586685062695222915686079286, −6.79361453462578930735208049385, −6.33216253885192386789698847118, −5.50026434999261826500281032618, −4.81757959105548043931199212153, −3.78467417888652900297326908736, −2.92771695708718701913079013959, −1.50901832098783487554330992976, −1.01887044403537953994741943704, 0,
1.01887044403537953994741943704, 1.50901832098783487554330992976, 2.92771695708718701913079013959, 3.78467417888652900297326908736, 4.81757959105548043931199212153, 5.50026434999261826500281032618, 6.33216253885192386789698847118, 6.79361453462578930735208049385, 7.25586685062695222915686079286
