| L(s) = 1 | + 3·3-s + 2·5-s + 9·9-s − 52·11-s − 86·13-s + 6·15-s + 30·17-s − 4·19-s − 120·23-s − 121·25-s + 27·27-s + 246·29-s + 80·31-s − 156·33-s − 290·37-s − 258·39-s + 374·41-s − 164·43-s + 18·45-s + 464·47-s + 90·51-s − 162·53-s − 104·55-s − 12·57-s + 180·59-s + 666·61-s − 172·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.178·5-s + 1/3·9-s − 1.42·11-s − 1.83·13-s + 0.103·15-s + 0.428·17-s − 0.0482·19-s − 1.08·23-s − 0.967·25-s + 0.192·27-s + 1.57·29-s + 0.463·31-s − 0.822·33-s − 1.28·37-s − 1.05·39-s + 1.42·41-s − 0.581·43-s + 0.0596·45-s + 1.44·47-s + 0.247·51-s − 0.419·53-s − 0.254·55-s − 0.0278·57-s + 0.397·59-s + 1.39·61-s − 0.328·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.878612156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.878612156\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 52 T + p^{3} T^{2} \) |
| 13 | \( 1 + 86 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 246 T + p^{3} T^{2} \) |
| 31 | \( 1 - 80 T + p^{3} T^{2} \) |
| 37 | \( 1 + 290 T + p^{3} T^{2} \) |
| 41 | \( 1 - 374 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 464 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 180 T + p^{3} T^{2} \) |
| 61 | \( 1 - 666 T + p^{3} T^{2} \) |
| 67 | \( 1 - 628 T + p^{3} T^{2} \) |
| 71 | \( 1 + 296 T + p^{3} T^{2} \) |
| 73 | \( 1 - 518 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1184 T + p^{3} T^{2} \) |
| 83 | \( 1 - 220 T + p^{3} T^{2} \) |
| 89 | \( 1 - 774 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1086 T + p^{3} T^{2} \) |
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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
−8.497088208076011658339545963350, −7.80978319505069333605468049562, −7.38056706274922705692169696798, −6.33810758881830641367262714759, −5.32039481063336279933762428694, −4.76169455513531175387137530231, −3.68257952374032695273112103135, −2.54354216554882508175328124800, −2.19916325956211072425624475806, −0.55841704968717829174852483660,
0.55841704968717829174852483660, 2.19916325956211072425624475806, 2.54354216554882508175328124800, 3.68257952374032695273112103135, 4.76169455513531175387137530231, 5.32039481063336279933762428694, 6.33810758881830641367262714759, 7.38056706274922705692169696798, 7.80978319505069333605468049562, 8.497088208076011658339545963350
