| L(s) = 1 | + 6.53·3-s + 20.9·5-s − 8.55·7-s + 15.6·9-s − 10.8·11-s + 54.7·13-s + 136.·15-s − 106.·17-s + 113.·19-s − 55.8·21-s + 112.·23-s + 312.·25-s − 74.1·27-s + 29·29-s + 102.·31-s − 70.5·33-s − 179.·35-s − 105.·37-s + 357.·39-s + 216.·41-s + 102.·43-s + 327.·45-s − 455.·47-s − 269.·49-s − 693.·51-s − 593.·53-s − 226.·55-s + ⋯ |
| L(s) = 1 | + 1.25·3-s + 1.87·5-s − 0.462·7-s + 0.579·9-s − 0.296·11-s + 1.16·13-s + 2.35·15-s − 1.51·17-s + 1.37·19-s − 0.580·21-s + 1.02·23-s + 2.50·25-s − 0.528·27-s + 0.185·29-s + 0.595·31-s − 0.372·33-s − 0.864·35-s − 0.469·37-s + 1.46·39-s + 0.826·41-s + 0.362·43-s + 1.08·45-s − 1.41·47-s − 0.786·49-s − 1.90·51-s − 1.53·53-s − 0.554·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.096863260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.096863260\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 - 6.53T + 27T^{2} \) |
| 5 | \( 1 - 20.9T + 125T^{2} \) |
| 7 | \( 1 + 8.55T + 343T^{2} \) |
| 11 | \( 1 + 10.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 54.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 112.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 102.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 105.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 216.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 102.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 455.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 593.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 558.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 473.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 193.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 2.38T + 3.57e5T^{2} \) |
| 73 | \( 1 - 119.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 964.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 772.T + 7.04e5T^{2} \) |
| 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
−10.38780668676736015304150279861, −9.347367088583399736087221698301, −9.172026583339415878381122827313, −8.159959328803664759465415490319, −6.80735412743129240754385357893, −6.05689925061209784022294754103, −4.93728733085446509153356845336, −3.29953706281125415385028687802, −2.51649734489822270078173874144, −1.41334968019554810185361494025,
1.41334968019554810185361494025, 2.51649734489822270078173874144, 3.29953706281125415385028687802, 4.93728733085446509153356845336, 6.05689925061209784022294754103, 6.80735412743129240754385357893, 8.159959328803664759465415490319, 9.172026583339415878381122827313, 9.347367088583399736087221698301, 10.38780668676736015304150279861
