| L(s) = 1 | + 2.74·3-s + 3.77·5-s + 2.22·7-s + 4.54·9-s − 0.380·13-s + 10.3·15-s − 3.15·17-s − 19-s + 6.11·21-s + 7.90·23-s + 9.22·25-s + 4.25·27-s + 5.59·29-s − 3.96·31-s + 8.40·35-s + 3.34·37-s − 1.04·39-s − 0.889·41-s − 8.46·43-s + 17.1·45-s − 3.67·47-s − 2.03·49-s − 8.67·51-s − 3.93·53-s − 2.74·57-s + 11.0·59-s + 3.65·61-s + ⋯ |
| L(s) = 1 | + 1.58·3-s + 1.68·5-s + 0.841·7-s + 1.51·9-s − 0.105·13-s + 2.67·15-s − 0.765·17-s − 0.229·19-s + 1.33·21-s + 1.64·23-s + 1.84·25-s + 0.818·27-s + 1.03·29-s − 0.711·31-s + 1.42·35-s + 0.550·37-s − 0.167·39-s − 0.138·41-s − 1.29·43-s + 2.55·45-s − 0.535·47-s − 0.291·49-s − 1.21·51-s − 0.540·53-s − 0.363·57-s + 1.43·59-s + 0.467·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.359350279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.359350279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.74T + 3T^{2} \) |
| 5 | \( 1 - 3.77T + 5T^{2} \) |
| 7 | \( 1 - 2.22T + 7T^{2} \) |
| 13 | \( 1 + 0.380T + 13T^{2} \) |
| 17 | \( 1 + 3.15T + 17T^{2} \) |
| 23 | \( 1 - 7.90T + 23T^{2} \) |
| 29 | \( 1 - 5.59T + 29T^{2} \) |
| 31 | \( 1 + 3.96T + 31T^{2} \) |
| 37 | \( 1 - 3.34T + 37T^{2} \) |
| 41 | \( 1 + 0.889T + 41T^{2} \) |
| 43 | \( 1 + 8.46T + 43T^{2} \) |
| 47 | \( 1 + 3.67T + 47T^{2} \) |
| 53 | \( 1 + 3.93T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 3.65T + 61T^{2} \) |
| 67 | \( 1 - 6.20T + 67T^{2} \) |
| 71 | \( 1 - 0.936T + 71T^{2} \) |
| 73 | \( 1 + 6.15T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 9.43T + 89T^{2} \) |
| 97 | \( 1 + 0.252T + 97T^{2} \) |
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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
−7.983684158002400648924638820969, −6.87891992994048795512271173583, −6.68765373178044645991306864085, −5.50431714950552015813196172562, −4.97767625921620834846331265845, −4.20453037994023675395290340062, −3.12365431968958579966715075698, −2.55753853962685482497504991573, −1.89514336849928055213932642352, −1.27104225946315378946476584209,
1.27104225946315378946476584209, 1.89514336849928055213932642352, 2.55753853962685482497504991573, 3.12365431968958579966715075698, 4.20453037994023675395290340062, 4.97767625921620834846331265845, 5.50431714950552015813196172562, 6.68765373178044645991306864085, 6.87891992994048795512271173583, 7.983684158002400648924638820969
