| L(s) = 1 | + 2.18·3-s + 0.879·5-s + 1.65·7-s + 1.77·9-s + 3.53·11-s + 4.94·13-s + 1.92·15-s + 2.59·17-s + 3.61·21-s + 4.82·23-s − 4.22·25-s − 2.68·27-s + 6.87·29-s + 2.04·31-s + 7.71·33-s + 1.45·35-s + 5.18·37-s + 10.8·39-s − 1.73·41-s − 7.98·43-s + 1.55·45-s − 2.92·47-s − 4.26·49-s + 5.67·51-s − 12.6·53-s + 3.10·55-s − 7.58·59-s + ⋯ |
| L(s) = 1 | + 1.26·3-s + 0.393·5-s + 0.624·7-s + 0.591·9-s + 1.06·11-s + 1.37·13-s + 0.496·15-s + 0.629·17-s + 0.787·21-s + 1.00·23-s − 0.845·25-s − 0.515·27-s + 1.27·29-s + 0.366·31-s + 1.34·33-s + 0.245·35-s + 0.852·37-s + 1.72·39-s − 0.271·41-s − 1.21·43-s + 0.232·45-s − 0.426·47-s − 0.609·49-s + 0.794·51-s − 1.74·53-s + 0.418·55-s − 0.987·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.492908168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.492908168\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.18T + 3T^{2} \) |
| 5 | \( 1 - 0.879T + 5T^{2} \) |
| 7 | \( 1 - 1.65T + 7T^{2} \) |
| 11 | \( 1 - 3.53T + 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 - 2.59T + 17T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 - 6.87T + 29T^{2} \) |
| 31 | \( 1 - 2.04T + 31T^{2} \) |
| 37 | \( 1 - 5.18T + 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 + 7.98T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 7.58T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 16.0T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 4.73T + 73T^{2} \) |
| 79 | \( 1 + 1.21T + 79T^{2} \) |
| 83 | \( 1 - 5.87T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.103075977946390103126490838731, −7.77611391610238112048103740232, −6.56888357167837186702442168271, −6.21103501985920429356365650583, −5.13367150448770522250418545391, −4.31126849558707409729636320861, −3.47071545291507248965979556239, −2.95540816156261460261336271026, −1.77213410835076282026669399932, −1.22526169561082748677089854492,
1.22526169561082748677089854492, 1.77213410835076282026669399932, 2.95540816156261460261336271026, 3.47071545291507248965979556239, 4.31126849558707409729636320861, 5.13367150448770522250418545391, 6.21103501985920429356365650583, 6.56888357167837186702442168271, 7.77611391610238112048103740232, 8.103075977946390103126490838731
