| L(s) = 1 | − 4.23·2-s + 9.91·4-s + 1.28·5-s − 27.4·7-s − 8.11·8-s − 5.45·10-s − 25.7·11-s − 78.2·13-s + 116.·14-s − 44.9·16-s + 88.8·17-s + 39.5·19-s + 12.7·20-s + 108.·22-s + 122.·23-s − 123.·25-s + 331.·26-s − 272.·28-s − 289.·29-s + 90.5·31-s + 255.·32-s − 376.·34-s − 35.4·35-s − 122.·37-s − 167.·38-s − 10.4·40-s − 188.·41-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.23·4-s + 0.115·5-s − 1.48·7-s − 0.358·8-s − 0.172·10-s − 0.704·11-s − 1.66·13-s + 2.22·14-s − 0.702·16-s + 1.26·17-s + 0.477·19-s + 0.143·20-s + 1.05·22-s + 1.11·23-s − 0.986·25-s + 2.49·26-s − 1.84·28-s − 1.85·29-s + 0.524·31-s + 1.41·32-s − 1.89·34-s − 0.171·35-s − 0.543·37-s − 0.714·38-s − 0.0413·40-s − 0.719·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1081573233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1081573233\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 4.23T + 8T^{2} \) |
| 5 | \( 1 - 1.28T + 125T^{2} \) |
| 7 | \( 1 + 27.4T + 343T^{2} \) |
| 11 | \( 1 + 25.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 78.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 88.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 289.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 90.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 122.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 188.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 191.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 308.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 285.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 497.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 434.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 859.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 783.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 642.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 694.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.66e3T + 9.12e5T^{2} \) |
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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
−9.016960989881488789109748599274, −7.81052713366600919966687648235, −7.46317996975058371254160175127, −6.75502994324451653522588544277, −5.72536153064254718101396662343, −4.88497644541795156320026836194, −3.40187256826510435325137164526, −2.68647065577879858530665355915, −1.55915667056278072641434979457, −0.18518358457317012933571417693,
0.18518358457317012933571417693, 1.55915667056278072641434979457, 2.68647065577879858530665355915, 3.40187256826510435325137164526, 4.88497644541795156320026836194, 5.72536153064254718101396662343, 6.75502994324451653522588544277, 7.46317996975058371254160175127, 7.81052713366600919966687648235, 9.016960989881488789109748599274
