L(s) = 1 | − 2.32·3-s − 1.34·5-s + 2.42·9-s + 0.563·11-s + 4.61·13-s + 3.12·15-s + 1.38·17-s + 2.28·19-s − 23-s − 3.20·25-s + 1.33·27-s − 0.714·29-s + 7.27·31-s − 1.31·33-s + 3.33·37-s − 10.7·39-s + 5.66·41-s + 2.57·43-s − 3.25·45-s + 4.96·47-s − 3.23·51-s − 8.82·53-s − 0.754·55-s − 5.31·57-s + 5.96·59-s − 2.37·61-s − 6.18·65-s + ⋯ |
L(s) = 1 | − 1.34·3-s − 0.599·5-s + 0.808·9-s + 0.169·11-s + 1.28·13-s + 0.806·15-s + 0.336·17-s + 0.523·19-s − 0.208·23-s − 0.640·25-s + 0.256·27-s − 0.132·29-s + 1.30·31-s − 0.228·33-s + 0.547·37-s − 1.72·39-s + 0.884·41-s + 0.393·43-s − 0.485·45-s + 0.724·47-s − 0.452·51-s − 1.21·53-s − 0.101·55-s − 0.704·57-s + 0.777·59-s − 0.304·61-s − 0.767·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.113908334\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.113908334\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 2.32T + 3T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 11 | \( 1 - 0.563T + 11T^{2} \) |
| 13 | \( 1 - 4.61T + 13T^{2} \) |
| 17 | \( 1 - 1.38T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 29 | \( 1 + 0.714T + 29T^{2} \) |
| 31 | \( 1 - 7.27T + 31T^{2} \) |
| 37 | \( 1 - 3.33T + 37T^{2} \) |
| 41 | \( 1 - 5.66T + 41T^{2} \) |
| 43 | \( 1 - 2.57T + 43T^{2} \) |
| 47 | \( 1 - 4.96T + 47T^{2} \) |
| 53 | \( 1 + 8.82T + 53T^{2} \) |
| 59 | \( 1 - 5.96T + 59T^{2} \) |
| 61 | \( 1 + 2.37T + 61T^{2} \) |
| 67 | \( 1 - 0.900T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 - 1.05T + 73T^{2} \) |
| 79 | \( 1 - 2.07T + 79T^{2} \) |
| 83 | \( 1 - 8.53T + 83T^{2} \) |
| 89 | \( 1 - 4.65T + 89T^{2} \) |
| 97 | \( 1 + 4.50T + 97T^{2} \) |
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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
−7.80466721404491187796878585825, −6.85053837445621657340706924980, −6.26288016769085063621299988462, −5.77367985509895537223236075320, −5.08103553767046282728854704508, −4.24185134322929893110538614623, −3.70553715215140553736514137104, −2.68985167962151166436789920087, −1.33785736990279616306194960245, −0.61693432223475350994709300454,
0.61693432223475350994709300454, 1.33785736990279616306194960245, 2.68985167962151166436789920087, 3.70553715215140553736514137104, 4.24185134322929893110538614623, 5.08103553767046282728854704508, 5.77367985509895537223236075320, 6.26288016769085063621299988462, 6.85053837445621657340706924980, 7.80466721404491187796878585825