| L(s) = 1 | + 2.29·3-s + 1.61·7-s + 2.26·9-s + 2.79·11-s + 4.88·13-s + 3.54·17-s + 2.79·19-s + 3.71·21-s + 23-s − 1.67·27-s − 3.36·29-s + 4.84·31-s + 6.41·33-s + 3.87·37-s + 11.2·39-s − 0.327·41-s + 5.38·43-s − 0.0121·47-s − 4.38·49-s + 8.13·51-s − 0.866·53-s + 6.41·57-s − 2.96·59-s − 7.29·61-s + 3.67·63-s − 6.94·67-s + 2.29·69-s + ⋯ |
| L(s) = 1 | + 1.32·3-s + 0.611·7-s + 0.756·9-s + 0.841·11-s + 1.35·13-s + 0.859·17-s + 0.640·19-s + 0.810·21-s + 0.208·23-s − 0.322·27-s − 0.624·29-s + 0.870·31-s + 1.11·33-s + 0.636·37-s + 1.79·39-s − 0.0512·41-s + 0.820·43-s − 0.00176·47-s − 0.626·49-s + 1.13·51-s − 0.118·53-s + 0.849·57-s − 0.386·59-s − 0.934·61-s + 0.462·63-s − 0.847·67-s + 0.276·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.708680197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.708680197\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.29T + 3T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 - 2.79T + 11T^{2} \) |
| 13 | \( 1 - 4.88T + 13T^{2} \) |
| 17 | \( 1 - 3.54T + 17T^{2} \) |
| 19 | \( 1 - 2.79T + 19T^{2} \) |
| 29 | \( 1 + 3.36T + 29T^{2} \) |
| 31 | \( 1 - 4.84T + 31T^{2} \) |
| 37 | \( 1 - 3.87T + 37T^{2} \) |
| 41 | \( 1 + 0.327T + 41T^{2} \) |
| 43 | \( 1 - 5.38T + 43T^{2} \) |
| 47 | \( 1 + 0.0121T + 47T^{2} \) |
| 53 | \( 1 + 0.866T + 53T^{2} \) |
| 59 | \( 1 + 2.96T + 59T^{2} \) |
| 61 | \( 1 + 7.29T + 61T^{2} \) |
| 67 | \( 1 + 6.94T + 67T^{2} \) |
| 71 | \( 1 - 2.16T + 71T^{2} \) |
| 73 | \( 1 + 6.02T + 73T^{2} \) |
| 79 | \( 1 + 6.50T + 79T^{2} \) |
| 83 | \( 1 - 7.88T + 83T^{2} \) |
| 89 | \( 1 + 9.71T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.77937121235591503504730320648, −7.36260973264409352740210097418, −6.30762162516807584481393999096, −5.79477409095287552194954347955, −4.78215971678239180669892594263, −3.98919421135213364453493128073, −3.41649488170895698554396807886, −2.76664383151041446222335690435, −1.68463023915951867593200101888, −1.10525584489992883692023784665,
1.10525584489992883692023784665, 1.68463023915951867593200101888, 2.76664383151041446222335690435, 3.41649488170895698554396807886, 3.98919421135213364453493128073, 4.78215971678239180669892594263, 5.79477409095287552194954347955, 6.30762162516807584481393999096, 7.36260973264409352740210097418, 7.77937121235591503504730320648
