| L(s) = 1 | − 5-s − 3·11-s − 4·13-s + 4·19-s − 8·23-s − 4·25-s + 3·29-s + 5·31-s + 8·37-s + 8·41-s + 6·43-s + 10·47-s − 9·53-s + 3·55-s − 5·59-s + 10·61-s + 4·65-s + 6·67-s − 10·71-s − 2·73-s + 11·79-s + 7·83-s − 18·89-s − 4·95-s + 17·97-s − 2·101-s + 11·107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.904·11-s − 1.10·13-s + 0.917·19-s − 1.66·23-s − 4/5·25-s + 0.557·29-s + 0.898·31-s + 1.31·37-s + 1.24·41-s + 0.914·43-s + 1.45·47-s − 1.23·53-s + 0.404·55-s − 0.650·59-s + 1.28·61-s + 0.496·65-s + 0.733·67-s − 1.18·71-s − 0.234·73-s + 1.23·79-s + 0.768·83-s − 1.90·89-s − 0.410·95-s + 1.72·97-s − 0.199·101-s + 1.06·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.251073185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.251073185\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−8.360451959638380886764948992387, −7.69460985079885241062719833772, −7.43884217549794085426971367853, −6.22611532765628300070427835172, −5.60267032534603783427998213305, −4.66740916943855638629991224884, −4.03863894567815535080980917834, −2.88475291513598245872161177124, −2.20706871286241338218264018898, −0.63422137058927113139572540422,
0.63422137058927113139572540422, 2.20706871286241338218264018898, 2.88475291513598245872161177124, 4.03863894567815535080980917834, 4.66740916943855638629991224884, 5.60267032534603783427998213305, 6.22611532765628300070427835172, 7.43884217549794085426971367853, 7.69460985079885241062719833772, 8.360451959638380886764948992387
