L(s) = 1 | + 2·2-s + 2·4-s + 4·5-s − 4·7-s + 8·10-s + 6·11-s − 4·13-s − 8·14-s − 4·16-s + 4·17-s − 7·19-s + 8·20-s + 12·22-s + 6·23-s + 11·25-s − 8·26-s − 8·28-s + 6·29-s − 2·31-s − 8·32-s + 8·34-s − 16·35-s − 14·38-s − 8·43-s + 12·44-s + 12·46-s + 9·47-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s + 1.78·5-s − 1.51·7-s + 2.52·10-s + 1.80·11-s − 1.10·13-s − 2.13·14-s − 16-s + 0.970·17-s − 1.60·19-s + 1.78·20-s + 2.55·22-s + 1.25·23-s + 11/5·25-s − 1.56·26-s − 1.51·28-s + 1.11·29-s − 0.359·31-s − 1.41·32-s + 1.37·34-s − 2.70·35-s − 2.27·38-s − 1.21·43-s + 1.80·44-s + 1.76·46-s + 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.838359529\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.838359529\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5077 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.56126870755163, −14.05021953982237, −13.56693925040079, −13.13829038993687, −12.73489120161547, −12.17879003079911, −12.01103688269687, −11.05885342406502, −10.27744978639475, −10.02539817947612, −9.408108433754635, −9.010798890653609, −8.614233567333854, −7.137554480062958, −6.772480740250183, −6.507090595799147, −5.880905547949235, −5.555077531192093, −4.798388036170816, −4.225828609152225, −3.523633469085458, −2.913656662784745, −2.442106717102284, −1.691708795020791, −0.7194774100373959,
0.7194774100373959, 1.691708795020791, 2.442106717102284, 2.913656662784745, 3.523633469085458, 4.225828609152225, 4.798388036170816, 5.555077531192093, 5.880905547949235, 6.507090595799147, 6.772480740250183, 7.137554480062958, 8.614233567333854, 9.010798890653609, 9.408108433754635, 10.02539817947612, 10.27744978639475, 11.05885342406502, 12.01103688269687, 12.17879003079911, 12.73489120161547, 13.13829038993687, 13.56693925040079, 14.05021953982237, 14.56126870755163