| L(s) = 1 | − 2.20·2-s − 3.13·4-s − 7·7-s + 24.5·8-s − 56.2·11-s − 38.9·13-s + 15.4·14-s − 29.1·16-s − 119.·17-s − 13.0·19-s + 124.·22-s − 130.·23-s + 85.8·26-s + 21.9·28-s − 77.9·29-s + 61.0·31-s − 132.·32-s + 263.·34-s − 167.·37-s + 28.6·38-s − 436.·41-s + 393.·43-s + 175.·44-s + 288.·46-s + 365.·47-s + 49·49-s + 121.·52-s + ⋯ |
| L(s) = 1 | − 0.780·2-s − 0.391·4-s − 0.377·7-s + 1.08·8-s − 1.54·11-s − 0.829·13-s + 0.294·14-s − 0.455·16-s − 1.70·17-s − 0.157·19-s + 1.20·22-s − 1.18·23-s + 0.647·26-s + 0.147·28-s − 0.499·29-s + 0.353·31-s − 0.730·32-s + 1.32·34-s − 0.743·37-s + 0.122·38-s − 1.66·41-s + 1.39·43-s + 0.602·44-s + 0.923·46-s + 1.13·47-s + 0.142·49-s + 0.324·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.04705919951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04705919951\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 2.20T + 8T^{2} \) |
| 11 | \( 1 + 56.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 13.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 77.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 61.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 436.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 393.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 365.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 282.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 414.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 563.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 395.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 103.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 128.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 641.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 512.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 186.T + 9.12e5T^{2} \) |
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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
−9.078443779515860506008944706500, −8.324300581526122361559481375494, −7.64488266331319713582989413144, −6.91473161786412866489015871260, −5.77198535805669700472821867756, −4.85343629662081405566866655788, −4.15069047471810470902152820333, −2.75418151606096743764519593709, −1.85151882981594851685028708603, −0.11257698843409200745545831660,
0.11257698843409200745545831660, 1.85151882981594851685028708603, 2.75418151606096743764519593709, 4.15069047471810470902152820333, 4.85343629662081405566866655788, 5.77198535805669700472821867756, 6.91473161786412866489015871260, 7.64488266331319713582989413144, 8.324300581526122361559481375494, 9.078443779515860506008944706500
