| L(s) = 1 | + 1.60·3-s − 1.38·5-s + 0.488·7-s − 0.439·9-s + 11-s + 4.26·13-s − 2.21·15-s + 3.59·17-s + 4.15·19-s + 0.781·21-s − 0.836·23-s − 3.08·25-s − 5.50·27-s + 0.335·29-s + 1.28·31-s + 1.60·33-s − 0.675·35-s − 5.32·37-s + 6.82·39-s − 0.0625·41-s + 6.89·43-s + 0.607·45-s − 2.70·47-s − 6.76·49-s + 5.76·51-s + 11.7·53-s − 1.38·55-s + ⋯ |
| L(s) = 1 | + 0.923·3-s − 0.618·5-s + 0.184·7-s − 0.146·9-s + 0.301·11-s + 1.18·13-s − 0.571·15-s + 0.873·17-s + 0.953·19-s + 0.170·21-s − 0.174·23-s − 0.617·25-s − 1.05·27-s + 0.0623·29-s + 0.230·31-s + 0.278·33-s − 0.114·35-s − 0.876·37-s + 1.09·39-s − 0.00977·41-s + 1.05·43-s + 0.0905·45-s − 0.394·47-s − 0.965·49-s + 0.806·51-s + 1.61·53-s − 0.186·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.716928623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.716928623\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 - 1.60T + 3T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 7 | \( 1 - 0.488T + 7T^{2} \) |
| 13 | \( 1 - 4.26T + 13T^{2} \) |
| 17 | \( 1 - 3.59T + 17T^{2} \) |
| 19 | \( 1 - 4.15T + 19T^{2} \) |
| 23 | \( 1 + 0.836T + 23T^{2} \) |
| 29 | \( 1 - 0.335T + 29T^{2} \) |
| 31 | \( 1 - 1.28T + 31T^{2} \) |
| 37 | \( 1 + 5.32T + 37T^{2} \) |
| 41 | \( 1 + 0.0625T + 41T^{2} \) |
| 43 | \( 1 - 6.89T + 43T^{2} \) |
| 47 | \( 1 + 2.70T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 8.67T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 0.352T + 67T^{2} \) |
| 71 | \( 1 + 7.36T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 3.25T + 89T^{2} \) |
| 97 | \( 1 + 19.5T + 97T^{2} \) |
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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
−8.254154449593832700836261211246, −7.54786871031699986453957095991, −6.84863820158832905273824795429, −5.84321902246066892471587866659, −5.31001745102346164511845175626, −4.07288159429429893403096647237, −3.63552158627903933603214686235, −2.96463400352681566818140778508, −1.90815451685282882621101133182, −0.851296726361600646090315391002,
0.851296726361600646090315391002, 1.90815451685282882621101133182, 2.96463400352681566818140778508, 3.63552158627903933603214686235, 4.07288159429429893403096647237, 5.31001745102346164511845175626, 5.84321902246066892471587866659, 6.84863820158832905273824795429, 7.54786871031699986453957095991, 8.254154449593832700836261211246
