L(s) = 1 | − 2-s + 4-s − 3·5-s + 7-s − 8-s + 3·10-s + 6·11-s − 14-s + 16-s + 3·17-s − 2·19-s − 3·20-s − 6·22-s + 4·25-s + 28-s − 6·29-s + 4·31-s − 32-s − 3·34-s − 3·35-s + 7·37-s + 2·38-s + 3·40-s − 43-s + 6·44-s + 3·47-s − 6·49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s + 1.80·11-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.458·19-s − 0.670·20-s − 1.27·22-s + 4/5·25-s + 0.188·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s − 0.507·35-s + 1.15·37-s + 0.324·38-s + 0.474·40-s − 0.152·43-s + 0.904·44-s + 0.437·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.114281696\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.114281696\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.699198430344978985948613366569, −7.934997178452817067223303679813, −7.46023947905323814388126308639, −6.63349412690092297262196593118, −5.90864573169190576818907822580, −4.60999990936908806684732327864, −3.94348155625524089488702113109, −3.19596255080634681974520626766, −1.76973285518057250387253289317, −0.74226904844910325205307127476,
0.74226904844910325205307127476, 1.76973285518057250387253289317, 3.19596255080634681974520626766, 3.94348155625524089488702113109, 4.60999990936908806684732327864, 5.90864573169190576818907822580, 6.63349412690092297262196593118, 7.46023947905323814388126308639, 7.934997178452817067223303679813, 8.699198430344978985948613366569