L(s) = 1 | − 1.41·2-s − 3-s + 1.41·6-s + 2.82·8-s + 9-s + 3.41·11-s − 1.58·13-s − 4.00·16-s − 6.24·17-s − 1.41·18-s + 6.65·19-s − 4.82·22-s + 6.24·23-s − 2.82·24-s + 2.24·26-s − 27-s − 0.242·29-s + 0.171·31-s − 3.41·33-s + 8.82·34-s + 5.58·37-s − 9.41·38-s + 1.58·39-s − 2.24·41-s + 10.4·43-s + ⋯ |
L(s) = 1 | − 1.00·2-s − 0.577·3-s + 0.577·6-s + 0.999·8-s + 0.333·9-s + 1.02·11-s − 0.439·13-s − 1.00·16-s − 1.51·17-s − 0.333·18-s + 1.52·19-s − 1.02·22-s + 1.30·23-s − 0.577·24-s + 0.439·26-s − 0.192·27-s − 0.0450·29-s + 0.0308·31-s − 0.594·33-s + 1.51·34-s + 0.918·37-s − 1.52·38-s + 0.253·39-s − 0.350·41-s + 1.58·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8053243814\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8053243814\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 - 6.65T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 + 0.242T + 29T^{2} \) |
| 31 | \( 1 - 0.171T + 31T^{2} \) |
| 37 | \( 1 - 5.58T + 37T^{2} \) |
| 41 | \( 1 + 2.24T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 9.31T + 47T^{2} \) |
| 53 | \( 1 + 1.17T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 3.41T + 71T^{2} \) |
| 73 | \( 1 - 2.07T + 73T^{2} \) |
| 79 | \( 1 + 4.65T + 79T^{2} \) |
| 83 | \( 1 - 5.41T + 83T^{2} \) |
| 89 | \( 1 - 3.75T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.863701418106296579101451158336, −7.69947008765459759289671968342, −7.22598224244007192603762208176, −6.52319003675254118251816681949, −5.57062530098527140489646703832, −4.66235134942954129317589692882, −4.13391911713437633549715854212, −2.81403547287180611764521706507, −1.54299918890633481153414445335, −0.67992057839945025960631793873,
0.67992057839945025960631793873, 1.54299918890633481153414445335, 2.81403547287180611764521706507, 4.13391911713437633549715854212, 4.66235134942954129317589692882, 5.57062530098527140489646703832, 6.52319003675254118251816681949, 7.22598224244007192603762208176, 7.69947008765459759289671968342, 8.863701418106296579101451158336