| L(s) = 1 | − 3·3-s + 9·9-s − 4·11-s − 54·13-s − 114·17-s − 44·19-s + 96·23-s − 27·27-s + 134·29-s + 272·31-s + 12·33-s + 98·37-s + 162·39-s − 6·41-s + 12·43-s − 200·47-s − 343·49-s + 342·51-s − 654·53-s + 132·57-s − 36·59-s − 442·61-s − 188·67-s − 288·69-s + 632·71-s + 390·73-s − 688·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.109·11-s − 1.15·13-s − 1.62·17-s − 0.531·19-s + 0.870·23-s − 0.192·27-s + 0.858·29-s + 1.57·31-s + 0.0633·33-s + 0.435·37-s + 0.665·39-s − 0.0228·41-s + 0.0425·43-s − 0.620·47-s − 49-s + 0.939·51-s − 1.69·53-s + 0.306·57-s − 0.0794·59-s − 0.927·61-s − 0.342·67-s − 0.502·69-s + 1.05·71-s + 0.625·73-s − 0.979·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.127673459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.127673459\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 - 134 T + p^{3} T^{2} \) |
| 31 | \( 1 - 272 T + p^{3} T^{2} \) |
| 37 | \( 1 - 98 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 - 12 T + p^{3} T^{2} \) |
| 47 | \( 1 + 200 T + p^{3} T^{2} \) |
| 53 | \( 1 + 654 T + p^{3} T^{2} \) |
| 59 | \( 1 + 36 T + p^{3} T^{2} \) |
| 61 | \( 1 + 442 T + p^{3} T^{2} \) |
| 67 | \( 1 + 188 T + p^{3} T^{2} \) |
| 71 | \( 1 - 632 T + p^{3} T^{2} \) |
| 73 | \( 1 - 390 T + p^{3} T^{2} \) |
| 79 | \( 1 + 688 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 694 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1726 T + p^{3} T^{2} \) |
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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
−9.462133564375155757780830919589, −8.576710005825834251960503169871, −7.67567492962458383437305013109, −6.69986355353428014340868339669, −6.22828235956972809939572296565, −4.74721620426036817622192712655, −4.66521744061865218319592594254, −3.03998691763911519954704372807, −2.01839085830507140682592217102, −0.54299681418069313559570091520,
0.54299681418069313559570091520, 2.01839085830507140682592217102, 3.03998691763911519954704372807, 4.66521744061865218319592594254, 4.74721620426036817622192712655, 6.22828235956972809939572296565, 6.69986355353428014340868339669, 7.67567492962458383437305013109, 8.576710005825834251960503169871, 9.462133564375155757780830919589
