| L(s) = 1 | + 4·5-s + 7-s − 3·11-s − 13-s + 7·17-s + 4·19-s + 2·23-s + 11·25-s + 2·29-s + 4·35-s + 7·37-s − 3·41-s − 5·43-s + 12·47-s − 6·49-s + 8·53-s − 12·55-s + 59-s − 14·61-s − 4·65-s + 4·67-s − 15·71-s − 4·73-s − 3·77-s − 5·79-s + 83-s + 28·85-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.377·7-s − 0.904·11-s − 0.277·13-s + 1.69·17-s + 0.917·19-s + 0.417·23-s + 11/5·25-s + 0.371·29-s + 0.676·35-s + 1.15·37-s − 0.468·41-s − 0.762·43-s + 1.75·47-s − 6/7·49-s + 1.09·53-s − 1.61·55-s + 0.130·59-s − 1.79·61-s − 0.496·65-s + 0.488·67-s − 1.78·71-s − 0.468·73-s − 0.341·77-s − 0.562·79-s + 0.109·83-s + 3.03·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.482090532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.482090532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.69118000359950473425923332880, −7.17840654334004396064982102537, −6.21486061483213025559173636804, −5.56871916417724600084565138693, −5.30026474556613426659499237234, −4.49139720644722643857153181283, −3.13458661498625466734140414171, −2.69518412451525491078036836923, −1.71373971078533193861408913167, −0.980899788697466720612744179575,
0.980899788697466720612744179575, 1.71373971078533193861408913167, 2.69518412451525491078036836923, 3.13458661498625466734140414171, 4.49139720644722643857153181283, 5.30026474556613426659499237234, 5.56871916417724600084565138693, 6.21486061483213025559173636804, 7.17840654334004396064982102537, 7.69118000359950473425923332880
