| L(s) = 1 | − 2-s − 2.72·3-s + 4-s + 0.0314·5-s + 2.72·6-s + 1.42·7-s − 8-s + 4.43·9-s − 0.0314·10-s − 2.88·11-s − 2.72·12-s − 4.12·13-s − 1.42·14-s − 0.0857·15-s + 16-s + 7.89·17-s − 4.43·18-s + 5.87·19-s + 0.0314·20-s − 3.88·21-s + 2.88·22-s − 9.03·23-s + 2.72·24-s − 4.99·25-s + 4.12·26-s − 3.90·27-s + 1.42·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.57·3-s + 0.5·4-s + 0.0140·5-s + 1.11·6-s + 0.538·7-s − 0.353·8-s + 1.47·9-s − 0.00994·10-s − 0.869·11-s − 0.787·12-s − 1.14·13-s − 0.381·14-s − 0.0221·15-s + 0.250·16-s + 1.91·17-s − 1.04·18-s + 1.34·19-s + 0.00703·20-s − 0.848·21-s + 0.614·22-s − 1.88·23-s + 0.556·24-s − 0.999·25-s + 0.808·26-s − 0.752·27-s + 0.269·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + 2.72T + 3T^{2} \) |
| 5 | \( 1 - 0.0314T + 5T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 + 2.88T + 11T^{2} \) |
| 13 | \( 1 + 4.12T + 13T^{2} \) |
| 17 | \( 1 - 7.89T + 17T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 23 | \( 1 + 9.03T + 23T^{2} \) |
| 29 | \( 1 - 3.11T + 29T^{2} \) |
| 31 | \( 1 + 3.90T + 31T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 + 1.61T + 43T^{2} \) |
| 47 | \( 1 + 0.804T + 47T^{2} \) |
| 53 | \( 1 - 7.26T + 53T^{2} \) |
| 59 | \( 1 - 8.95T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 2.69T + 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 + 9.69T + 73T^{2} \) |
| 79 | \( 1 + 4.63T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 - 6.43T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−8.132447205926131895984846029455, −7.69329365800447897152007161741, −7.08657261440059648633943392459, −5.93304929748236127726865513383, −5.48599292055821365866674803051, −4.90916840650082089562510382782, −3.64422145535836974674819933230, −2.30388111378530375418099677048, −1.13097760096021804900424940686, 0,
1.13097760096021804900424940686, 2.30388111378530375418099677048, 3.64422145535836974674819933230, 4.90916840650082089562510382782, 5.48599292055821365866674803051, 5.93304929748236127726865513383, 7.08657261440059648633943392459, 7.69329365800447897152007161741, 8.132447205926131895984846029455
