| L(s) = 1 | + 0.353·3-s + 5-s − 4.30·7-s − 2.87·9-s + 6.01·11-s + 5.95·13-s + 0.353·15-s + 3.87·17-s − 1.52·21-s + 0.783·23-s + 25-s − 2.07·27-s − 7.96·29-s + 4.49·31-s + 2.12·33-s − 4.30·35-s + 0.988·37-s + 2.10·39-s − 6.30·41-s − 1.57·43-s − 2.87·45-s − 1.26·47-s + 11.5·49-s + 1.36·51-s + 8.14·53-s + 6.01·55-s − 5.25·59-s + ⋯ |
| L(s) = 1 | + 0.204·3-s + 0.447·5-s − 1.62·7-s − 0.958·9-s + 1.81·11-s + 1.65·13-s + 0.0912·15-s + 0.939·17-s − 0.332·21-s + 0.163·23-s + 0.200·25-s − 0.399·27-s − 1.47·29-s + 0.806·31-s + 0.369·33-s − 0.727·35-s + 0.162·37-s + 0.336·39-s − 0.984·41-s − 0.239·43-s − 0.428·45-s − 0.184·47-s + 1.64·49-s + 0.191·51-s + 1.11·53-s + 0.810·55-s − 0.684·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.183438602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.183438602\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.353T + 3T^{2} \) |
| 7 | \( 1 + 4.30T + 7T^{2} \) |
| 11 | \( 1 - 6.01T + 11T^{2} \) |
| 13 | \( 1 - 5.95T + 13T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 23 | \( 1 - 0.783T + 23T^{2} \) |
| 29 | \( 1 + 7.96T + 29T^{2} \) |
| 31 | \( 1 - 4.49T + 31T^{2} \) |
| 37 | \( 1 - 0.988T + 37T^{2} \) |
| 41 | \( 1 + 6.30T + 41T^{2} \) |
| 43 | \( 1 + 1.57T + 43T^{2} \) |
| 47 | \( 1 + 1.26T + 47T^{2} \) |
| 53 | \( 1 - 8.14T + 53T^{2} \) |
| 59 | \( 1 + 5.25T + 59T^{2} \) |
| 61 | \( 1 - 5.61T + 61T^{2} \) |
| 67 | \( 1 + 7.04T + 67T^{2} \) |
| 71 | \( 1 - 5.81T + 71T^{2} \) |
| 73 | \( 1 + 9.24T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 6.58T + 83T^{2} \) |
| 89 | \( 1 - 3.38T + 89T^{2} \) |
| 97 | \( 1 - 7.39T + 97T^{2} \) |
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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
−8.031613602042909854889368413866, −7.00133162818842773521592936932, −6.32044268608327410831879925176, −6.09903508008338671970671769800, −5.35663754159239295941950437407, −3.94134610229572868564715399020, −3.55108032989879135995386120437, −2.96602370539487319254466840046, −1.73029756235817197108940398902, −0.76185950175554343871135212223,
0.76185950175554343871135212223, 1.73029756235817197108940398902, 2.96602370539487319254466840046, 3.55108032989879135995386120437, 3.94134610229572868564715399020, 5.35663754159239295941950437407, 6.09903508008338671970671769800, 6.32044268608327410831879925176, 7.00133162818842773521592936932, 8.031613602042909854889368413866
