| L(s) = 1 | + 9.40·3-s + 5·5-s − 32.3·7-s + 61.4·9-s − 31.5·11-s + 13·13-s + 47.0·15-s + 70.9·17-s + 128.·19-s − 304.·21-s + 114.·23-s + 25·25-s + 323.·27-s + 56.3·29-s − 260.·31-s − 296.·33-s − 161.·35-s + 386.·37-s + 122.·39-s − 82.8·41-s + 457.·43-s + 307.·45-s − 226.·47-s + 703.·49-s + 666.·51-s + 212.·53-s − 157.·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s + 0.447·5-s − 1.74·7-s + 2.27·9-s − 0.863·11-s + 0.277·13-s + 0.809·15-s + 1.01·17-s + 1.54·19-s − 3.16·21-s + 1.04·23-s + 0.200·25-s + 2.30·27-s + 0.361·29-s − 1.51·31-s − 1.56·33-s − 0.781·35-s + 1.71·37-s + 0.501·39-s − 0.315·41-s + 1.62·43-s + 1.01·45-s − 0.701·47-s + 2.04·49-s + 1.83·51-s + 0.550·53-s − 0.386·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.067405122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.067405122\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 13 | \( 1 - 13T \) |
| good | 3 | \( 1 - 9.40T + 27T^{2} \) |
| 7 | \( 1 + 32.3T + 343T^{2} \) |
| 11 | \( 1 + 31.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 70.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 128.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 56.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 260.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 386.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 82.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 457.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 226.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 212.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 39.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 726.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 453.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 890.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 544.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 927.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 50.9T + 5.71e5T^{2} \) |
| 89 | \( 1 - 702.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 555.T + 9.12e5T^{2} \) |
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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
−9.444831991364799944341557583617, −9.007747266769069557232598314609, −7.77543298331073702238674276683, −7.36683068950093447193266953186, −6.28538232668349506872941069836, −5.21977619517837198075286108336, −3.71874528427823887148281574312, −3.11736282639827301715660569132, −2.50963178380182078701162028122, −1.01089677666303805017606766073,
1.01089677666303805017606766073, 2.50963178380182078701162028122, 3.11736282639827301715660569132, 3.71874528427823887148281574312, 5.21977619517837198075286108336, 6.28538232668349506872941069836, 7.36683068950093447193266953186, 7.77543298331073702238674276683, 9.007747266769069557232598314609, 9.444831991364799944341557583617
