| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 5.65·11-s + 2·13-s + 15-s + 7.65·17-s + 5.65·19-s + 21-s + 5.65·23-s + 25-s + 27-s − 7.65·29-s + 4·31-s − 5.65·33-s + 35-s + 0.343·37-s + 2·39-s + 2·41-s − 9.65·43-s + 45-s − 13.6·47-s + 49-s + 7.65·51-s + 7.65·53-s − 5.65·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s − 1.70·11-s + 0.554·13-s + 0.258·15-s + 1.85·17-s + 1.29·19-s + 0.218·21-s + 1.17·23-s + 0.200·25-s + 0.192·27-s − 1.42·29-s + 0.718·31-s − 0.984·33-s + 0.169·35-s + 0.0564·37-s + 0.320·39-s + 0.312·41-s − 1.47·43-s + 0.149·45-s − 1.99·47-s + 0.142·49-s + 1.07·51-s + 1.05·53-s − 0.762·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.152705508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.152705508\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 0.343T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 + 13.6T + 47T^{2} \) |
| 53 | \( 1 - 7.65T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−10.01554870998532599790967156885, −9.544709100434793646921321311081, −8.291530514995535253565657600115, −7.85655760809085391360077214915, −6.94551819290591796102252533663, −5.46214366715280708381153673448, −5.17722853416948962235825545177, −3.52239563815426301983598487010, −2.73268581061183372205264944327, −1.31914919422026613728919459094,
1.31914919422026613728919459094, 2.73268581061183372205264944327, 3.52239563815426301983598487010, 5.17722853416948962235825545177, 5.46214366715280708381153673448, 6.94551819290591796102252533663, 7.85655760809085391360077214915, 8.291530514995535253565657600115, 9.544709100434793646921321311081, 10.01554870998532599790967156885
