| L(s) = 1 | + 3.15·3-s − 5·5-s − 11.8·7-s − 17.0·9-s + 63.8·11-s + 11.1·13-s − 15.7·15-s − 3.42·17-s + 20.8·19-s − 37.4·21-s + 23·23-s + 25·25-s − 139.·27-s − 264.·29-s − 129.·31-s + 201.·33-s + 59.2·35-s + 248.·37-s + 35.1·39-s − 285.·41-s − 213.·43-s + 85.1·45-s + 627.·47-s − 202.·49-s − 10.8·51-s + 402.·53-s − 319.·55-s + ⋯ |
| L(s) = 1 | + 0.607·3-s − 0.447·5-s − 0.639·7-s − 0.630·9-s + 1.75·11-s + 0.237·13-s − 0.271·15-s − 0.0488·17-s + 0.251·19-s − 0.388·21-s + 0.208·23-s + 0.200·25-s − 0.990·27-s − 1.69·29-s − 0.750·31-s + 1.06·33-s + 0.286·35-s + 1.10·37-s + 0.144·39-s − 1.08·41-s − 0.757·43-s + 0.282·45-s + 1.94·47-s − 0.590·49-s − 0.0296·51-s + 1.04·53-s − 0.782·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 3.15T + 27T^{2} \) |
| 7 | \( 1 + 11.8T + 343T^{2} \) |
| 11 | \( 1 - 63.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 3.42T + 4.91e3T^{2} \) |
| 19 | \( 1 - 20.8T + 6.85e3T^{2} \) |
| 29 | \( 1 + 264.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 248.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 285.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 213.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 627.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 402.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 525.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 632.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 21.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 878.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 453.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 631.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 653.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 333.T + 9.12e5T^{2} \) |
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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
−8.678259709207397633952518301768, −7.74464188650878320247077819178, −6.94566213735165420577567765575, −6.20160370925268533935683762956, −5.31787916009501679482316929220, −3.85865955033681512594683080904, −3.67877085116585660949338010205, −2.52367835088160206855465445893, −1.31737291477286692634115403659, 0,
1.31737291477286692634115403659, 2.52367835088160206855465445893, 3.67877085116585660949338010205, 3.85865955033681512594683080904, 5.31787916009501679482316929220, 6.20160370925268533935683762956, 6.94566213735165420577567765575, 7.74464188650878320247077819178, 8.678259709207397633952518301768
