| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 6·6-s + 7·7-s + 8·8-s − 18·9-s − 17·11-s + 12·12-s + 81·13-s + 14·14-s + 16·16-s + 91·17-s − 36·18-s + 102·19-s + 21·21-s − 34·22-s + 90·23-s + 24·24-s + 162·26-s − 135·27-s + 28·28-s − 129·29-s + 116·31-s + 32·32-s − 51·33-s + 182·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.465·11-s + 0.288·12-s + 1.72·13-s + 0.267·14-s + 1/4·16-s + 1.29·17-s − 0.471·18-s + 1.23·19-s + 0.218·21-s − 0.329·22-s + 0.815·23-s + 0.204·24-s + 1.22·26-s − 0.962·27-s + 0.188·28-s − 0.826·29-s + 0.672·31-s + 0.176·32-s − 0.269·33-s + 0.918·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.836548327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.836548327\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 - p T + p^{3} T^{2} \) |
| 11 | \( 1 + 17 T + p^{3} T^{2} \) |
| 13 | \( 1 - 81 T + p^{3} T^{2} \) |
| 17 | \( 1 - 91 T + p^{3} T^{2} \) |
| 19 | \( 1 - 102 T + p^{3} T^{2} \) |
| 23 | \( 1 - 90 T + p^{3} T^{2} \) |
| 29 | \( 1 + 129 T + p^{3} T^{2} \) |
| 31 | \( 1 - 116 T + p^{3} T^{2} \) |
| 37 | \( 1 + 314 T + p^{3} T^{2} \) |
| 41 | \( 1 + 124 T + p^{3} T^{2} \) |
| 43 | \( 1 - 434 T + p^{3} T^{2} \) |
| 47 | \( 1 + 497 T + p^{3} T^{2} \) |
| 53 | \( 1 - 584 T + p^{3} T^{2} \) |
| 59 | \( 1 + 332 T + p^{3} T^{2} \) |
| 61 | \( 1 - 220 T + p^{3} T^{2} \) |
| 67 | \( 1 + 384 T + p^{3} T^{2} \) |
| 71 | \( 1 + 664 T + p^{3} T^{2} \) |
| 73 | \( 1 + 230 T + p^{3} T^{2} \) |
| 79 | \( 1 - 361 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1172 T + p^{3} T^{2} \) |
| 89 | \( 1 - 40 T + p^{3} T^{2} \) |
| 97 | \( 1 - 175 T + p^{3} T^{2} \) |
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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
−11.24422458006338640259965305251, −10.30190921307101676004324180549, −9.041651081003043836967506553921, −8.195503568782016239868104639674, −7.33138644589751668624576694014, −5.92263838864270130467279825728, −5.23665124191037791885969380687, −3.69103271319213818006082075871, −2.95550254771160321732249006504, −1.33295321208164163523978045249,
1.33295321208164163523978045249, 2.95550254771160321732249006504, 3.69103271319213818006082075871, 5.23665124191037791885969380687, 5.92263838864270130467279825728, 7.33138644589751668624576694014, 8.195503568782016239868104639674, 9.041651081003043836967506553921, 10.30190921307101676004324180549, 11.24422458006338640259965305251
