| L(s) = 1 | + 5.61·3-s + 13.8·5-s − 9.22·7-s + 4.56·9-s − 59.8·11-s − 38.2·13-s + 77.6·15-s − 47.0·17-s + 19·19-s − 51.8·21-s + 179.·23-s + 66.1·25-s − 126.·27-s + 96.6·29-s + 125.·31-s − 336.·33-s − 127.·35-s − 407.·37-s − 214.·39-s − 220.·41-s − 100.·43-s + 63.0·45-s − 213.·47-s − 257.·49-s − 264.·51-s − 19.8·53-s − 827.·55-s + ⋯ |
| L(s) = 1 | + 1.08·3-s + 1.23·5-s − 0.497·7-s + 0.169·9-s − 1.64·11-s − 0.815·13-s + 1.33·15-s − 0.671·17-s + 0.229·19-s − 0.538·21-s + 1.62·23-s + 0.529·25-s − 0.898·27-s + 0.618·29-s + 0.729·31-s − 1.77·33-s − 0.615·35-s − 1.81·37-s − 0.881·39-s − 0.840·41-s − 0.357·43-s + 0.209·45-s − 0.663·47-s − 0.752·49-s − 0.726·51-s − 0.0513·53-s − 2.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 5.61T + 27T^{2} \) |
| 5 | \( 1 - 13.8T + 125T^{2} \) |
| 7 | \( 1 + 9.22T + 343T^{2} \) |
| 11 | \( 1 + 59.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 47.0T + 4.91e3T^{2} \) |
| 23 | \( 1 - 179.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 96.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 125.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 407.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 220.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 100.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 213.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 19.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 97.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 266.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 627.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 644.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 807.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 785.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 234.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.34e3T + 9.12e5T^{2} \) |
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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
−8.942142816211152271668764029669, −8.322120178589551716424563689467, −7.34763157858977993218573059202, −6.54716213425966633017891131726, −5.42541513982929421876956168481, −4.83430006546242323052663889032, −3.14010192441708057902545619881, −2.70947680707328080312257889876, −1.79728691094932214265185777940, 0,
1.79728691094932214265185777940, 2.70947680707328080312257889876, 3.14010192441708057902545619881, 4.83430006546242323052663889032, 5.42541513982929421876956168481, 6.54716213425966633017891131726, 7.34763157858977993218573059202, 8.322120178589551716424563689467, 8.942142816211152271668764029669
