| L(s) = 1 | − 2-s + 4-s − 5-s + 2·7-s − 8-s + 10-s + 2·11-s − 2·14-s + 16-s − 5·17-s + 2·19-s − 20-s − 2·22-s − 6·23-s − 4·25-s + 2·28-s + 9·29-s + 4·31-s − 32-s + 5·34-s − 2·35-s + 11·37-s − 2·38-s + 40-s + 5·41-s + 10·43-s + 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.603·11-s − 0.534·14-s + 1/4·16-s − 1.21·17-s + 0.458·19-s − 0.223·20-s − 0.426·22-s − 1.25·23-s − 4/5·25-s + 0.377·28-s + 1.67·29-s + 0.718·31-s − 0.176·32-s + 0.857·34-s − 0.338·35-s + 1.80·37-s − 0.324·38-s + 0.158·40-s + 0.780·41-s + 1.52·43-s + 0.301·44-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
| \(L(1)\) |
\(\approx\) |
\(1.277109990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.277109990\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.718271935584440339873451757405, −7.86716293168731238112023995708, −7.59570311850396464656844334232, −6.42392713155869381940807788029, −5.98855118952487149300267164409, −4.59265124932082688708507660637, −4.18540850693176983023428564757, −2.86657729430232147791994970248, −1.92419283425109288512088509081, −0.77534609559859883840518392730,
0.77534609559859883840518392730, 1.92419283425109288512088509081, 2.86657729430232147791994970248, 4.18540850693176983023428564757, 4.59265124932082688708507660637, 5.98855118952487149300267164409, 6.42392713155869381940807788029, 7.59570311850396464656844334232, 7.86716293168731238112023995708, 8.718271935584440339873451757405
