| L(s) = 1 | − 3·3-s + 5·5-s + 32·7-s + 9·9-s + 36·11-s − 10·13-s − 15·15-s − 78·17-s + 140·19-s − 96·21-s − 192·23-s + 25·25-s − 27·27-s + 6·29-s − 16·31-s − 108·33-s + 160·35-s − 34·37-s + 30·39-s − 390·41-s − 52·43-s + 45·45-s + 408·47-s + 681·49-s + 234·51-s − 114·53-s + 180·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.72·7-s + 1/3·9-s + 0.986·11-s − 0.213·13-s − 0.258·15-s − 1.11·17-s + 1.69·19-s − 0.997·21-s − 1.74·23-s + 1/5·25-s − 0.192·27-s + 0.0384·29-s − 0.0926·31-s − 0.569·33-s + 0.772·35-s − 0.151·37-s + 0.123·39-s − 1.48·41-s − 0.184·43-s + 0.149·45-s + 1.26·47-s + 1.98·49-s + 0.642·51-s − 0.295·53-s + 0.441·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.475579723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.475579723\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 10 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 - 140 T + p^{3} T^{2} \) |
| 23 | \( 1 + 192 T + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 + 16 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 390 T + p^{3} T^{2} \) |
| 43 | \( 1 + 52 T + p^{3} T^{2} \) |
| 47 | \( 1 - 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 114 T + p^{3} T^{2} \) |
| 59 | \( 1 - 516 T + p^{3} T^{2} \) |
| 61 | \( 1 + 58 T + p^{3} T^{2} \) |
| 67 | \( 1 + 892 T + p^{3} T^{2} \) |
| 71 | \( 1 + 120 T + p^{3} T^{2} \) |
| 73 | \( 1 + 646 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1168 T + p^{3} T^{2} \) |
| 83 | \( 1 + 732 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1590 T + p^{3} T^{2} \) |
| 97 | \( 1 - 2 p T + p^{3} T^{2} \) |
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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
−14.43200596551567343832241203107, −13.70491554573411682214832490416, −11.95294009791997255865367742390, −11.41166455792298060058744728844, −10.07605105350119559510509162368, −8.661615715626482662230767080891, −7.25696192972094874036790233744, −5.69308505956036567627196751081, −4.43322963900143531387926850092, −1.63497771165194779201344719277,
1.63497771165194779201344719277, 4.43322963900143531387926850092, 5.69308505956036567627196751081, 7.25696192972094874036790233744, 8.661615715626482662230767080891, 10.07605105350119559510509162368, 11.41166455792298060058744728844, 11.95294009791997255865367742390, 13.70491554573411682214832490416, 14.43200596551567343832241203107
