| L(s) = 1 | + 4.75·2-s + 14.5·4-s − 7·7-s + 31.3·8-s − 7.31·11-s + 4.15·13-s − 33.2·14-s + 32.2·16-s + 53.5·17-s + 88.9·19-s − 34.7·22-s + 156.·23-s + 19.7·26-s − 102.·28-s − 42.2·29-s − 14.0·31-s − 97.4·32-s + 254.·34-s + 293.·37-s + 422.·38-s + 127.·41-s + 210.·43-s − 106.·44-s + 745.·46-s − 468.·47-s + 49·49-s + 60.6·52-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 1.82·4-s − 0.377·7-s + 1.38·8-s − 0.200·11-s + 0.0886·13-s − 0.635·14-s + 0.504·16-s + 0.763·17-s + 1.07·19-s − 0.337·22-s + 1.42·23-s + 0.148·26-s − 0.689·28-s − 0.270·29-s − 0.0812·31-s − 0.538·32-s + 1.28·34-s + 1.30·37-s + 1.80·38-s + 0.484·41-s + 0.745·43-s − 0.365·44-s + 2.38·46-s − 1.45·47-s + 0.142·49-s + 0.161·52-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
| \(L(2)\) |
\(\approx\) |
\(6.431937189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.431937189\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4.75T + 8T^{2} \) |
| 11 | \( 1 + 7.31T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4.15T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 156.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 42.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 14.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 293.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 127.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 210.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 468.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 115.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 314.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 768.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 717.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 737.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 477.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 279.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 776.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 29.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 231.T + 9.12e5T^{2} \) |
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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
−9.191955115831193195707336342639, −7.964238557234693713171959587048, −7.15279522305684363713462416163, −6.43046326335053846875588594829, −5.51109324368772818803354104563, −5.04252034791946493649606683482, −3.95733600105687281412030861443, −3.22514177416966815855907842392, −2.45150477552762655548252988549, −0.981774935826628107407346253521,
0.981774935826628107407346253521, 2.45150477552762655548252988549, 3.22514177416966815855907842392, 3.95733600105687281412030861443, 5.04252034791946493649606683482, 5.51109324368772818803354104563, 6.43046326335053846875588594829, 7.15279522305684363713462416163, 7.964238557234693713171959587048, 9.191955115831193195707336342639
