| L(s) = 1 | − 5-s + 9·7-s − 17·11-s + 44·13-s + 56·17-s − 94·19-s + 50·23-s − 124·25-s + 30·29-s + 139·31-s − 9·35-s + 174·37-s + 318·41-s − 242·43-s + 630·47-s − 262·49-s − 547·53-s + 17·55-s − 236·59-s − 328·61-s − 44·65-s + 614·67-s − 296·71-s + 433·73-s − 153·77-s + 56·79-s − 1.22e3·83-s + ⋯ |
| L(s) = 1 | − 0.0894·5-s + 0.485·7-s − 0.465·11-s + 0.938·13-s + 0.798·17-s − 1.13·19-s + 0.453·23-s − 0.991·25-s + 0.192·29-s + 0.805·31-s − 0.0434·35-s + 0.773·37-s + 1.21·41-s − 0.858·43-s + 1.95·47-s − 0.763·49-s − 1.41·53-s + 0.0416·55-s − 0.520·59-s − 0.688·61-s − 0.0839·65-s + 1.11·67-s − 0.494·71-s + 0.694·73-s − 0.226·77-s + 0.0797·79-s − 1.62·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.254578312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.254578312\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 - 9 T + p^{3} T^{2} \) |
| 11 | \( 1 + 17 T + p^{3} T^{2} \) |
| 13 | \( 1 - 44 T + p^{3} T^{2} \) |
| 17 | \( 1 - 56 T + p^{3} T^{2} \) |
| 19 | \( 1 + 94 T + p^{3} T^{2} \) |
| 23 | \( 1 - 50 T + p^{3} T^{2} \) |
| 29 | \( 1 - 30 T + p^{3} T^{2} \) |
| 31 | \( 1 - 139 T + p^{3} T^{2} \) |
| 37 | \( 1 - 174 T + p^{3} T^{2} \) |
| 41 | \( 1 - 318 T + p^{3} T^{2} \) |
| 43 | \( 1 + 242 T + p^{3} T^{2} \) |
| 47 | \( 1 - 630 T + p^{3} T^{2} \) |
| 53 | \( 1 + 547 T + p^{3} T^{2} \) |
| 59 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 61 | \( 1 + 328 T + p^{3} T^{2} \) |
| 67 | \( 1 - 614 T + p^{3} T^{2} \) |
| 71 | \( 1 + 296 T + p^{3} T^{2} \) |
| 73 | \( 1 - 433 T + p^{3} T^{2} \) |
| 79 | \( 1 - 56 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1225 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1506 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1391 T + p^{3} T^{2} \) |
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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
−8.867702098867525133666652039395, −8.091418251271526788076414898607, −7.60536049498202445502558904501, −6.41775452208538138539272571620, −5.81995880824898951865698069403, −4.79251692106357293832839640725, −3.99844775888009405048899789313, −2.95754711943461076883940536528, −1.84926127163821232886193696579, −0.72550054947440261267752929175,
0.72550054947440261267752929175, 1.84926127163821232886193696579, 2.95754711943461076883940536528, 3.99844775888009405048899789313, 4.79251692106357293832839640725, 5.81995880824898951865698069403, 6.41775452208538138539272571620, 7.60536049498202445502558904501, 8.091418251271526788076414898607, 8.867702098867525133666652039395
