| L(s) = 1 | + 1.40·3-s − 1.57·7-s − 1.03·9-s + 4.35·11-s − 0.964·13-s + 0.300·17-s − 8.62·19-s − 2.20·21-s + 23-s − 5.65·27-s + 4.76·29-s − 5.59·31-s + 6.11·33-s + 4.38·37-s − 1.35·39-s − 6.62·41-s + 1.72·43-s + 0.687·47-s − 4.53·49-s + 0.421·51-s − 8.05·53-s − 12.1·57-s + 5.74·59-s − 13.6·61-s + 1.61·63-s − 6.49·67-s + 1.40·69-s + ⋯ |
| L(s) = 1 | + 0.810·3-s − 0.593·7-s − 0.343·9-s + 1.31·11-s − 0.267·13-s + 0.0728·17-s − 1.97·19-s − 0.481·21-s + 0.208·23-s − 1.08·27-s + 0.885·29-s − 1.00·31-s + 1.06·33-s + 0.720·37-s − 0.216·39-s − 1.03·41-s + 0.263·43-s + 0.100·47-s − 0.647·49-s + 0.0589·51-s − 1.10·53-s − 1.60·57-s + 0.748·59-s − 1.74·61-s + 0.204·63-s − 0.792·67-s + 0.168·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 1.40T + 3T^{2} \) |
| 7 | \( 1 + 1.57T + 7T^{2} \) |
| 11 | \( 1 - 4.35T + 11T^{2} \) |
| 13 | \( 1 + 0.964T + 13T^{2} \) |
| 17 | \( 1 - 0.300T + 17T^{2} \) |
| 19 | \( 1 + 8.62T + 19T^{2} \) |
| 29 | \( 1 - 4.76T + 29T^{2} \) |
| 31 | \( 1 + 5.59T + 31T^{2} \) |
| 37 | \( 1 - 4.38T + 37T^{2} \) |
| 41 | \( 1 + 6.62T + 41T^{2} \) |
| 43 | \( 1 - 1.72T + 43T^{2} \) |
| 47 | \( 1 - 0.687T + 47T^{2} \) |
| 53 | \( 1 + 8.05T + 53T^{2} \) |
| 59 | \( 1 - 5.74T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 6.49T + 67T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 + 6.35T + 73T^{2} \) |
| 79 | \( 1 + 6.95T + 79T^{2} \) |
| 83 | \( 1 - 0.185T + 83T^{2} \) |
| 89 | \( 1 + 1.64T + 89T^{2} \) |
| 97 | \( 1 - 9.21T + 97T^{2} \) |
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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
−8.127254688677773806596057523624, −7.24604474065048994407177819609, −6.42364594782590939769517426740, −6.06295129918597396634235856359, −4.81966066022030856862992321051, −4.02319152761844961646828011258, −3.33658368423770658637522308227, −2.50334991013751456543815305884, −1.58244811161969268585875473045, 0,
1.58244811161969268585875473045, 2.50334991013751456543815305884, 3.33658368423770658637522308227, 4.02319152761844961646828011258, 4.81966066022030856862992321051, 6.06295129918597396634235856359, 6.42364594782590939769517426740, 7.24604474065048994407177819609, 8.127254688677773806596057523624
