| L(s) = 1 | − 0.839·3-s − 5-s + 3.45·7-s − 2.29·9-s + 4.68·11-s + 1.95·13-s + 0.839·15-s + 5.62·17-s − 8.21·19-s − 2.90·21-s + 6.06·23-s + 25-s + 4.44·27-s + 1.07·29-s + 4.63·31-s − 3.93·33-s − 3.45·35-s + 7.20·37-s − 1.64·39-s − 4.22·41-s + 12.4·43-s + 2.29·45-s − 11.6·47-s + 4.93·49-s − 4.72·51-s + 3.48·53-s − 4.68·55-s + ⋯ |
| L(s) = 1 | − 0.484·3-s − 0.447·5-s + 1.30·7-s − 0.764·9-s + 1.41·11-s + 0.542·13-s + 0.216·15-s + 1.36·17-s − 1.88·19-s − 0.633·21-s + 1.26·23-s + 0.200·25-s + 0.855·27-s + 0.199·29-s + 0.832·31-s − 0.685·33-s − 0.583·35-s + 1.18·37-s − 0.262·39-s − 0.660·41-s + 1.89·43-s + 0.342·45-s − 1.69·47-s + 0.704·49-s − 0.661·51-s + 0.479·53-s − 0.631·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.121972845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.121972845\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 0.839T + 3T^{2} \) |
| 7 | \( 1 - 3.45T + 7T^{2} \) |
| 11 | \( 1 - 4.68T + 11T^{2} \) |
| 13 | \( 1 - 1.95T + 13T^{2} \) |
| 17 | \( 1 - 5.62T + 17T^{2} \) |
| 19 | \( 1 + 8.21T + 19T^{2} \) |
| 23 | \( 1 - 6.06T + 23T^{2} \) |
| 29 | \( 1 - 1.07T + 29T^{2} \) |
| 31 | \( 1 - 4.63T + 31T^{2} \) |
| 37 | \( 1 - 7.20T + 37T^{2} \) |
| 41 | \( 1 + 4.22T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 3.48T + 53T^{2} \) |
| 59 | \( 1 + 6.99T + 59T^{2} \) |
| 61 | \( 1 + 1.37T + 61T^{2} \) |
| 67 | \( 1 - 8.42T + 67T^{2} \) |
| 71 | \( 1 - 8.86T + 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 + 6.37T + 79T^{2} \) |
| 83 | \( 1 - 6.85T + 83T^{2} \) |
| 89 | \( 1 + 7.62T + 89T^{2} \) |
| 97 | \( 1 - 2.15T + 97T^{2} \) |
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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
−7.989475821992061575729140537144, −7.10682486795376276669238767671, −6.32187456120570277212332035484, −5.87697685174249848146274902989, −4.90627374251134742304317211635, −4.42697183765750230981993351757, −3.62987009559104163315732729230, −2.69583614399789343809281985497, −1.52166182253741066881027190896, −0.815779006328262711976330246054,
0.815779006328262711976330246054, 1.52166182253741066881027190896, 2.69583614399789343809281985497, 3.62987009559104163315732729230, 4.42697183765750230981993351757, 4.90627374251134742304317211635, 5.87697685174249848146274902989, 6.32187456120570277212332035484, 7.10682486795376276669238767671, 7.989475821992061575729140537144
