| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s − 5·11-s + 3·13-s − 4·14-s + 16-s + 7·17-s − 3·19-s − 5·22-s + 7·23-s + 3·26-s − 4·28-s + 9·29-s + 7·31-s + 32-s + 7·34-s + 5·37-s − 3·38-s − 8·41-s + 8·43-s − 5·44-s + 7·46-s + 8·47-s + 9·49-s + 3·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s − 1.50·11-s + 0.832·13-s − 1.06·14-s + 1/4·16-s + 1.69·17-s − 0.688·19-s − 1.06·22-s + 1.45·23-s + 0.588·26-s − 0.755·28-s + 1.67·29-s + 1.25·31-s + 0.176·32-s + 1.20·34-s + 0.821·37-s − 0.486·38-s − 1.24·41-s + 1.21·43-s − 0.753·44-s + 1.03·46-s + 1.16·47-s + 9/7·49-s + 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.731500041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.731500041\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.91794891121409, −13.47386806239102, −13.08441909017114, −12.63680784061232, −12.30412281310285, −11.73014607173765, −10.99872862962156, −10.49065328894574, −10.10436167326140, −9.823494584353057, −8.849777870993120, −8.537244572770848, −7.783133730714325, −7.347221373321476, −6.694396486511627, −6.169606325759488, −5.819706170811305, −5.150151530048970, −4.649915476397325, −3.854163699481283, −3.285712517614169, −2.780342708215634, −2.497436901243318, −1.156237953722435, −0.6313640729599269,
0.6313640729599269, 1.156237953722435, 2.497436901243318, 2.780342708215634, 3.285712517614169, 3.854163699481283, 4.649915476397325, 5.150151530048970, 5.819706170811305, 6.169606325759488, 6.694396486511627, 7.347221373321476, 7.783133730714325, 8.537244572770848, 8.849777870993120, 9.823494584353057, 10.10436167326140, 10.49065328894574, 10.99872862962156, 11.73014607173765, 12.30412281310285, 12.63680784061232, 13.08441909017114, 13.47386806239102, 13.91794891121409
