| L(s) = 1 | + 4·2-s + 8·4-s + 5·5-s + 6·7-s + 20·10-s − 32·11-s − 38·13-s + 24·14-s − 64·16-s − 26·17-s + 100·19-s + 40·20-s − 128·22-s + 78·23-s + 25·25-s − 152·26-s + 48·28-s + 50·29-s − 108·31-s − 256·32-s − 104·34-s + 30·35-s + 266·37-s + 400·38-s − 22·41-s + 442·43-s − 256·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s + 0.323·7-s + 0.632·10-s − 0.877·11-s − 0.810·13-s + 0.458·14-s − 16-s − 0.370·17-s + 1.20·19-s + 0.447·20-s − 1.24·22-s + 0.707·23-s + 1/5·25-s − 1.14·26-s + 0.323·28-s + 0.320·29-s − 0.625·31-s − 1.41·32-s − 0.524·34-s + 0.144·35-s + 1.18·37-s + 1.70·38-s − 0.0838·41-s + 1.56·43-s − 0.877·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.536676230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.536676230\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 2 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 7 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 32 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 26 T + p^{3} T^{2} \) |
| 19 | \( 1 - 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 78 T + p^{3} T^{2} \) |
| 29 | \( 1 - 50 T + p^{3} T^{2} \) |
| 31 | \( 1 + 108 T + p^{3} T^{2} \) |
| 37 | \( 1 - 266 T + p^{3} T^{2} \) |
| 41 | \( 1 + 22 T + p^{3} T^{2} \) |
| 43 | \( 1 - 442 T + p^{3} T^{2} \) |
| 47 | \( 1 - 514 T + p^{3} T^{2} \) |
| 53 | \( 1 + 2 T + p^{3} T^{2} \) |
| 59 | \( 1 + 500 T + p^{3} T^{2} \) |
| 61 | \( 1 + 518 T + p^{3} T^{2} \) |
| 67 | \( 1 - 126 T + p^{3} T^{2} \) |
| 71 | \( 1 + 412 T + p^{3} T^{2} \) |
| 73 | \( 1 + 878 T + p^{3} T^{2} \) |
| 79 | \( 1 - 600 T + p^{3} T^{2} \) |
| 83 | \( 1 + 282 T + p^{3} T^{2} \) |
| 89 | \( 1 - 150 T + p^{3} T^{2} \) |
| 97 | \( 1 - 386 T + p^{3} T^{2} \) |
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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
−15.02210984640626865507790060331, −14.07637810416215704339053328492, −13.16870125563977638744407814213, −12.18469650940908618231665179716, −10.90702313512267444200924440980, −9.327272798277878694630788064100, −7.40775204093622856232137515033, −5.75386185154896584819978577710, −4.68855375702425266914192974411, −2.75409226362896290546787147957,
2.75409226362896290546787147957, 4.68855375702425266914192974411, 5.75386185154896584819978577710, 7.40775204093622856232137515033, 9.327272798277878694630788064100, 10.90702313512267444200924440980, 12.18469650940908618231665179716, 13.16870125563977638744407814213, 14.07637810416215704339053328492, 15.02210984640626865507790060331
