| L(s) = 1 | + 3-s + 0.462·5-s + 7-s + 9-s + 0.398·11-s − 0.860·13-s + 0.462·15-s − 1.60·17-s − 6.11·19-s + 21-s − 23-s − 4.78·25-s + 27-s − 8.24·29-s + 1.32·31-s + 0.398·33-s + 0.462·35-s + 10.7·37-s − 0.860·39-s + 1.04·41-s + 2.86·43-s + 0.462·45-s − 2.27·47-s + 49-s − 1.60·51-s − 4.46·53-s + 0.184·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.206·5-s + 0.377·7-s + 0.333·9-s + 0.120·11-s − 0.238·13-s + 0.119·15-s − 0.388·17-s − 1.40·19-s + 0.218·21-s − 0.208·23-s − 0.957·25-s + 0.192·27-s − 1.53·29-s + 0.237·31-s + 0.0693·33-s + 0.0781·35-s + 1.77·37-s − 0.137·39-s + 0.163·41-s + 0.436·43-s + 0.0689·45-s − 0.332·47-s + 0.142·49-s − 0.224·51-s − 0.612·53-s + 0.0248·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 0.462T + 5T^{2} \) |
| 11 | \( 1 - 0.398T + 11T^{2} \) |
| 13 | \( 1 + 0.860T + 13T^{2} \) |
| 17 | \( 1 + 1.60T + 17T^{2} \) |
| 19 | \( 1 + 6.11T + 19T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 - 1.32T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 43 | \( 1 - 2.86T + 43T^{2} \) |
| 47 | \( 1 + 2.27T + 47T^{2} \) |
| 53 | \( 1 + 4.46T + 53T^{2} \) |
| 59 | \( 1 - 1.90T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 2.33T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 + 5.84T + 73T^{2} \) |
| 79 | \( 1 - 1.75T + 79T^{2} \) |
| 83 | \( 1 - 8.76T + 83T^{2} \) |
| 89 | \( 1 - 1.91T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−7.81903717399780833114245721494, −6.84395454977280152928313257696, −6.14064774303998535772603750802, −5.50379329424009880217617469404, −4.34586912968767802836295442907, −4.17124867134059138436755191754, −2.99906996575158976123689015467, −2.21122049168949333093031203549, −1.52748723371001040326515311827, 0,
1.52748723371001040326515311827, 2.21122049168949333093031203549, 2.99906996575158976123689015467, 4.17124867134059138436755191754, 4.34586912968767802836295442907, 5.50379329424009880217617469404, 6.14064774303998535772603750802, 6.84395454977280152928313257696, 7.81903717399780833114245721494
