| L(s) = 1 | − 1.84·3-s + 0.765·5-s + 4.46·7-s + 0.414·9-s − 1.84·11-s − 1.41·15-s + 6.24·19-s − 8.24·21-s + 4.90·23-s − 4.41·25-s + 4.77·27-s + 2.29·29-s − 5.54·31-s + 3.41·33-s + 3.41·35-s − 1.84·37-s − 9.68·41-s + 1.75·43-s + 0.317·45-s + 7.17·47-s + 12.8·49-s + 11.0·53-s − 1.41·55-s − 11.5·57-s + 11.8·59-s + 12.7·61-s + 1.84·63-s + ⋯ |
| L(s) = 1 | − 1.06·3-s + 0.342·5-s + 1.68·7-s + 0.138·9-s − 0.557·11-s − 0.365·15-s + 1.43·19-s − 1.79·21-s + 1.02·23-s − 0.882·25-s + 0.919·27-s + 0.426·29-s − 0.995·31-s + 0.594·33-s + 0.577·35-s − 0.303·37-s − 1.51·41-s + 0.267·43-s + 0.0472·45-s + 1.04·47-s + 1.84·49-s + 1.52·53-s − 0.190·55-s − 1.52·57-s + 1.54·59-s + 1.63·61-s + 0.232·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.675773526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.675773526\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 1.84T + 3T^{2} \) |
| 5 | \( 1 - 0.765T + 5T^{2} \) |
| 7 | \( 1 - 4.46T + 7T^{2} \) |
| 11 | \( 1 + 1.84T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 19 | \( 1 - 6.24T + 19T^{2} \) |
| 23 | \( 1 - 4.90T + 23T^{2} \) |
| 29 | \( 1 - 2.29T + 29T^{2} \) |
| 31 | \( 1 + 5.54T + 31T^{2} \) |
| 37 | \( 1 + 1.84T + 37T^{2} \) |
| 41 | \( 1 + 9.68T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 - 7.17T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 - 2.48T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 1.75T + 83T^{2} \) |
| 89 | \( 1 + 9.65T + 89T^{2} \) |
| 97 | \( 1 + 2.03T + 97T^{2} \) |
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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
−8.381392538728304562053701903872, −7.40209335800366754293256648628, −7.02243098004214166971256601877, −5.75705577396396635901798502593, −5.37597354197355939825160725165, −4.97568061670969597682909654305, −3.98239382871007066342529203189, −2.76234899385338930928018945050, −1.72774048859909418292019116353, −0.797181165270139808735821920722,
0.797181165270139808735821920722, 1.72774048859909418292019116353, 2.76234899385338930928018945050, 3.98239382871007066342529203189, 4.97568061670969597682909654305, 5.37597354197355939825160725165, 5.75705577396396635901798502593, 7.02243098004214166971256601877, 7.40209335800366754293256648628, 8.381392538728304562053701903872
