| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s − 6.41·11-s + 12-s − 13-s − 15-s + 16-s + 3.24·17-s + 18-s − 4.65·19-s − 20-s − 6.41·22-s − 2.17·23-s + 24-s − 4·25-s − 26-s + 27-s + 7.48·29-s − 30-s + 1.41·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.93·11-s + 0.288·12-s − 0.277·13-s − 0.258·15-s + 0.250·16-s + 0.786·17-s + 0.235·18-s − 1.06·19-s − 0.223·20-s − 1.36·22-s − 0.452·23-s + 0.204·24-s − 0.800·25-s − 0.196·26-s + 0.192·27-s + 1.38·29-s − 0.182·30-s + 0.254·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 + 6.41T + 11T^{2} \) |
| 17 | \( 1 - 3.24T + 17T^{2} \) |
| 19 | \( 1 + 4.65T + 19T^{2} \) |
| 23 | \( 1 + 2.17T + 23T^{2} \) |
| 29 | \( 1 - 7.48T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 8.41T + 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 + 8.65T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 + 5.75T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 - 2.24T + 79T^{2} \) |
| 83 | \( 1 + 3.65T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 7.31T + 97T^{2} \) |
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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
−8.147236032374330448322800267331, −7.46660151077778050334636802937, −6.69321178483256316094987064104, −5.73295017213657611597707207199, −5.00649849469277909624954076827, −4.32465467240853572498843441969, −3.34183576380681669974137440925, −2.71958445992661757566858906348, −1.80760691237883472289884600279, 0,
1.80760691237883472289884600279, 2.71958445992661757566858906348, 3.34183576380681669974137440925, 4.32465467240853572498843441969, 5.00649849469277909624954076827, 5.73295017213657611597707207199, 6.69321178483256316094987064104, 7.46660151077778050334636802937, 8.147236032374330448322800267331
