| L(s) = 1 | − 3-s + 9-s − 4·11-s − 2·13-s − 17-s + 4·19-s − 27-s + 10·29-s − 8·31-s + 4·33-s − 2·37-s + 2·39-s + 10·41-s − 12·43-s − 7·49-s + 51-s + 6·53-s − 4·57-s + 12·59-s + 10·61-s + 12·67-s − 10·73-s + 8·79-s + 81-s − 4·83-s − 10·87-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.242·17-s + 0.917·19-s − 0.192·27-s + 1.85·29-s − 1.43·31-s + 0.696·33-s − 0.328·37-s + 0.320·39-s + 1.56·41-s − 1.82·43-s − 49-s + 0.140·51-s + 0.824·53-s − 0.529·57-s + 1.56·59-s + 1.28·61-s + 1.46·67-s − 1.17·73-s + 0.900·79-s + 1/9·81-s − 0.439·83-s − 1.07·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.256666194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.256666194\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.07351726667125, −13.24616978211466, −13.02100227613983, −12.53556076204301, −11.94490948790575, −11.45842745486949, −11.10062126933135, −10.35085154227663, −10.08529100789822, −9.667367350785422, −8.882020617886504, −8.384138591224052, −7.822436582698544, −7.283255033676479, −6.852054566477193, −6.243170053286029, −5.528757815493697, −5.120424323557300, −4.797833447862090, −3.952236886116293, −3.328480459946207, −2.601795183409403, −2.110092305157748, −1.142932887979034, −0.4111586269358455,
0.4111586269358455, 1.142932887979034, 2.110092305157748, 2.601795183409403, 3.328480459946207, 3.952236886116293, 4.797833447862090, 5.120424323557300, 5.528757815493697, 6.243170053286029, 6.852054566477193, 7.283255033676479, 7.822436582698544, 8.384138591224052, 8.882020617886504, 9.667367350785422, 10.08529100789822, 10.35085154227663, 11.10062126933135, 11.45842745486949, 11.94490948790575, 12.53556076204301, 13.02100227613983, 13.24616978211466, 14.07351726667125
