| L(s) = 1 | + 2·2-s + 9·3-s + 4·4-s + 20·5-s + 18·6-s + 8·8-s + 54·9-s + 40·10-s − 52·11-s + 36·12-s − 43·13-s + 180·15-s + 16·16-s + 50·17-s + 108·18-s + 74·19-s + 80·20-s − 104·22-s − 23·23-s + 72·24-s + 275·25-s − 86·26-s + 243·27-s − 7·29-s + 360·30-s + 273·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.78·5-s + 1.22·6-s + 0.353·8-s + 2·9-s + 1.26·10-s − 1.42·11-s + 0.866·12-s − 0.917·13-s + 3.09·15-s + 1/4·16-s + 0.713·17-s + 1.41·18-s + 0.893·19-s + 0.894·20-s − 1.00·22-s − 0.208·23-s + 0.612·24-s + 11/5·25-s − 0.648·26-s + 1.73·27-s − 0.0448·29-s + 2.19·30-s + 1.58·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.03157132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.03157132\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + p T \) |
| good | 3 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 5 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 52 T + p^{3} T^{2} \) |
| 13 | \( 1 + 43 T + p^{3} T^{2} \) |
| 17 | \( 1 - 50 T + p^{3} T^{2} \) |
| 19 | \( 1 - 74 T + p^{3} T^{2} \) |
| 29 | \( 1 + 7 T + p^{3} T^{2} \) |
| 31 | \( 1 - 273 T + p^{3} T^{2} \) |
| 37 | \( 1 + 4 T + p^{3} T^{2} \) |
| 41 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 152 T + p^{3} T^{2} \) |
| 47 | \( 1 + 75 T + p^{3} T^{2} \) |
| 53 | \( 1 - 86 T + p^{3} T^{2} \) |
| 59 | \( 1 - 444 T + p^{3} T^{2} \) |
| 61 | \( 1 + 262 T + p^{3} T^{2} \) |
| 67 | \( 1 - 764 T + p^{3} T^{2} \) |
| 71 | \( 1 + 21 T + p^{3} T^{2} \) |
| 73 | \( 1 + 681 T + p^{3} T^{2} \) |
| 79 | \( 1 - 426 T + p^{3} T^{2} \) |
| 83 | \( 1 + 902 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1272 T + p^{3} T^{2} \) |
| 97 | \( 1 - 342 T + p^{3} T^{2} \) |
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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
−8.659854681389875957833450273372, −7.87972928665703203077477167603, −7.24975486140213180815870721441, −6.28927987981991552499363426356, −5.30068344221281925851851628527, −4.83406513841279021135075150618, −3.45046274537069994381178339315, −2.63019472605708041710583678003, −2.34838616454164099072425878016, −1.30435290499050517761385589433,
1.30435290499050517761385589433, 2.34838616454164099072425878016, 2.63019472605708041710583678003, 3.45046274537069994381178339315, 4.83406513841279021135075150618, 5.30068344221281925851851628527, 6.28927987981991552499363426356, 7.24975486140213180815870721441, 7.87972928665703203077477167603, 8.659854681389875957833450273372
