L(s) = 1 | − 0.414·2-s + 3-s − 1.82·4-s − 0.414·6-s + 1.58·8-s + 9-s − 2·11-s − 1.82·12-s + 2.58·13-s + 3·16-s − 2.24·17-s − 0.414·18-s + 2.82·19-s + 0.828·22-s + 7.65·23-s + 1.58·24-s − 1.07·26-s + 27-s − 6.82·29-s + 1.17·31-s − 4.41·32-s − 2·33-s + 0.928·34-s − 1.82·36-s + 4·37-s − 1.17·38-s + 2.58·39-s + ⋯ |
L(s) = 1 | − 0.292·2-s + 0.577·3-s − 0.914·4-s − 0.169·6-s + 0.560·8-s + 0.333·9-s − 0.603·11-s − 0.527·12-s + 0.717·13-s + 0.750·16-s − 0.543·17-s − 0.0976·18-s + 0.648·19-s + 0.176·22-s + 1.59·23-s + 0.323·24-s − 0.210·26-s + 0.192·27-s − 1.26·29-s + 0.210·31-s − 0.780·32-s − 0.348·33-s + 0.159·34-s − 0.304·36-s + 0.657·37-s − 0.190·38-s + 0.414·39-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
\(L(1)\) |
\(\approx\) |
\(1.539530862\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.539530862\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.414T + 2T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 1.17T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 9.31T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 7.31T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 2.58T + 97T^{2} \) |
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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
−8.518941975047728879078266854356, −7.999832316299798033765249200153, −7.27807335604176674189236749169, −6.39824163937052954811198460182, −5.27697974758996349470661069181, −4.82051074060462128697470921502, −3.75349692209605763541666174657, −3.15740738618990452251776045459, −1.90980432137990327391261256039, −0.75808932091979279254199746409,
0.75808932091979279254199746409, 1.90980432137990327391261256039, 3.15740738618990452251776045459, 3.75349692209605763541666174657, 4.82051074060462128697470921502, 5.27697974758996349470661069181, 6.39824163937052954811198460182, 7.27807335604176674189236749169, 7.999832316299798033765249200153, 8.518941975047728879078266854356