| L(s) = 1 | + 1.18·3-s − 0.696·5-s + 2.51·7-s − 1.59·9-s − 0.459·11-s + 3.40·13-s − 0.825·15-s + 5.32·17-s + 0.179·19-s + 2.98·21-s + 1.56·23-s − 4.51·25-s − 5.44·27-s + 5.66·29-s − 1.09·31-s − 0.544·33-s − 1.75·35-s + 7.02·37-s + 4.02·39-s + 6.41·41-s + 4.43·43-s + 1.11·45-s + 11.5·47-s − 0.659·49-s + 6.30·51-s − 12.6·53-s + 0.319·55-s + ⋯ |
| L(s) = 1 | + 0.684·3-s − 0.311·5-s + 0.951·7-s − 0.532·9-s − 0.138·11-s + 0.943·13-s − 0.213·15-s + 1.29·17-s + 0.0412·19-s + 0.651·21-s + 0.327·23-s − 0.902·25-s − 1.04·27-s + 1.05·29-s − 0.196·31-s − 0.0947·33-s − 0.296·35-s + 1.15·37-s + 0.645·39-s + 1.00·41-s + 0.676·43-s + 0.165·45-s + 1.68·47-s − 0.0942·49-s + 0.882·51-s − 1.73·53-s + 0.0431·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.252611196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.252611196\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 - 1.18T + 3T^{2} \) |
| 5 | \( 1 + 0.696T + 5T^{2} \) |
| 7 | \( 1 - 2.51T + 7T^{2} \) |
| 11 | \( 1 + 0.459T + 11T^{2} \) |
| 13 | \( 1 - 3.40T + 13T^{2} \) |
| 17 | \( 1 - 5.32T + 17T^{2} \) |
| 19 | \( 1 - 0.179T + 19T^{2} \) |
| 23 | \( 1 - 1.56T + 23T^{2} \) |
| 29 | \( 1 - 5.66T + 29T^{2} \) |
| 31 | \( 1 + 1.09T + 31T^{2} \) |
| 37 | \( 1 - 7.02T + 37T^{2} \) |
| 41 | \( 1 - 6.41T + 41T^{2} \) |
| 43 | \( 1 - 4.43T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 9.38T + 61T^{2} \) |
| 67 | \( 1 + 8.91T + 67T^{2} \) |
| 71 | \( 1 + 5.22T + 71T^{2} \) |
| 73 | \( 1 - 9.43T + 73T^{2} \) |
| 79 | \( 1 + 8.98T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 7.32T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−9.421902504861379216551622512515, −8.729187135762642568155946300072, −7.85546513818964081153241277978, −7.70038087129161272317143176981, −6.19864282344558486880096374776, −5.47768298505997449215912754058, −4.36908534392481144446376284883, −3.46411380811617530389493607529, −2.49133921643382342275439815806, −1.15582395808914764195834061500,
1.15582395808914764195834061500, 2.49133921643382342275439815806, 3.46411380811617530389493607529, 4.36908534392481144446376284883, 5.47768298505997449215912754058, 6.19864282344558486880096374776, 7.70038087129161272317143176981, 7.85546513818964081153241277978, 8.729187135762642568155946300072, 9.421902504861379216551622512515
