| L(s) = 1 | + 3·3-s + 5-s + 6·9-s + 2·11-s + 6·13-s + 3·15-s + 2·17-s − 9·23-s + 25-s + 9·27-s − 3·29-s + 2·31-s + 6·33-s − 8·37-s + 18·39-s + 5·41-s − 43-s + 6·45-s + 8·47-s + 6·51-s − 4·53-s + 2·55-s + 8·59-s − 7·61-s + 6·65-s + 3·67-s − 27·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 2·9-s + 0.603·11-s + 1.66·13-s + 0.774·15-s + 0.485·17-s − 1.87·23-s + 1/5·25-s + 1.73·27-s − 0.557·29-s + 0.359·31-s + 1.04·33-s − 1.31·37-s + 2.88·39-s + 0.780·41-s − 0.152·43-s + 0.894·45-s + 1.16·47-s + 0.840·51-s − 0.549·53-s + 0.269·55-s + 1.04·59-s − 0.896·61-s + 0.744·65-s + 0.366·67-s − 3.25·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.937524695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.937524695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.80562523762995, −15.46117361556143, −14.73932370560332, −14.17975065462915, −13.76230026262328, −13.63560123834870, −12.74386562020122, −12.31194541456828, −11.50857595247741, −10.75293681186930, −10.15087937438910, −9.616112268941134, −9.018816957455804, −8.629365991337945, −8.006388031609608, −7.579539953597814, −6.650583091145001, −6.159751186914054, −5.402154311450890, −4.309229186131071, −3.683318191782294, −3.437394739727868, −2.342277385507208, −1.830417081970310, −1.029699076602034,
1.029699076602034, 1.830417081970310, 2.342277385507208, 3.437394739727868, 3.683318191782294, 4.309229186131071, 5.402154311450890, 6.159751186914054, 6.650583091145001, 7.579539953597814, 8.006388031609608, 8.629365991337945, 9.018816957455804, 9.616112268941134, 10.15087937438910, 10.75293681186930, 11.50857595247741, 12.31194541456828, 12.74386562020122, 13.63560123834870, 13.76230026262328, 14.17975065462915, 14.73932370560332, 15.46117361556143, 15.80562523762995
