| L(s) = 1 | − 3.05·3-s − 1.91·5-s + 4.03·7-s + 6.34·9-s − 13-s + 5.85·15-s + 7.69·17-s − 1.94·19-s − 12.3·21-s − 1.72·23-s − 1.32·25-s − 10.2·27-s + 10.1·29-s + 9.09·31-s − 7.73·35-s + 11.8·37-s + 3.05·39-s + 5.28·41-s + 1.49·43-s − 12.1·45-s + 7.84·47-s + 9.28·49-s − 23.5·51-s − 10.5·53-s + 5.94·57-s − 1.12·59-s − 8.65·61-s + ⋯ |
| L(s) = 1 | − 1.76·3-s − 0.857·5-s + 1.52·7-s + 2.11·9-s − 0.277·13-s + 1.51·15-s + 1.86·17-s − 0.446·19-s − 2.69·21-s − 0.358·23-s − 0.264·25-s − 1.96·27-s + 1.89·29-s + 1.63·31-s − 1.30·35-s + 1.95·37-s + 0.489·39-s + 0.824·41-s + 0.227·43-s − 1.81·45-s + 1.14·47-s + 1.32·49-s − 3.29·51-s − 1.44·53-s + 0.787·57-s − 0.147·59-s − 1.10·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.232093844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.232093844\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 3.05T + 3T^{2} \) |
| 5 | \( 1 + 1.91T + 5T^{2} \) |
| 7 | \( 1 - 4.03T + 7T^{2} \) |
| 17 | \( 1 - 7.69T + 17T^{2} \) |
| 19 | \( 1 + 1.94T + 19T^{2} \) |
| 23 | \( 1 + 1.72T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 9.09T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 - 5.28T + 41T^{2} \) |
| 43 | \( 1 - 1.49T + 43T^{2} \) |
| 47 | \( 1 - 7.84T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 + 8.65T + 61T^{2} \) |
| 67 | \( 1 - 1.55T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 3.99T + 73T^{2} \) |
| 79 | \( 1 + 9.44T + 79T^{2} \) |
| 83 | \( 1 + 4.29T + 83T^{2} \) |
| 89 | \( 1 + 4.54T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.78434521516364547156060123243, −7.54066396114336774986377728061, −6.40299601251592166260170443336, −5.92228462188404180176435658201, −5.10296972988780791119648787665, −4.54314691354978725365702459557, −4.11824478495011675779179623394, −2.72840674060333231181782287206, −1.32311602555398247282087484310, −0.74482449745049906665944077716,
0.74482449745049906665944077716, 1.32311602555398247282087484310, 2.72840674060333231181782287206, 4.11824478495011675779179623394, 4.54314691354978725365702459557, 5.10296972988780791119648787665, 5.92228462188404180176435658201, 6.40299601251592166260170443336, 7.54066396114336774986377728061, 7.78434521516364547156060123243
