| L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s + 6·11-s − 13-s + 4·14-s + 16-s − 4·17-s + 2·19-s + 6·22-s − 6·23-s − 26-s + 4·28-s + 10·29-s + 4·31-s + 32-s − 4·34-s + 6·37-s + 2·38-s − 10·41-s + 6·44-s − 6·46-s + 8·47-s + 9·49-s − 52-s − 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 1.80·11-s − 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.970·17-s + 0.458·19-s + 1.27·22-s − 1.25·23-s − 0.196·26-s + 0.755·28-s + 1.85·29-s + 0.718·31-s + 0.176·32-s − 0.685·34-s + 0.986·37-s + 0.324·38-s − 1.56·41-s + 0.904·44-s − 0.884·46-s + 1.16·47-s + 9/7·49-s − 0.138·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.481041573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.481041573\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.267457766766554180261302096318, −7.21711567530854826784063836710, −6.62198806112113371536994919615, −5.97838242286196789678644755584, −5.03679342732839928114971332368, −4.38350774702154640897955577571, −4.00861134990217667206387117655, −2.79378338238435915309448129089, −1.86138661653525093628569663191, −1.11928205020771569067669460478,
1.11928205020771569067669460478, 1.86138661653525093628569663191, 2.79378338238435915309448129089, 4.00861134990217667206387117655, 4.38350774702154640897955577571, 5.03679342732839928114971332368, 5.97838242286196789678644755584, 6.62198806112113371536994919615, 7.21711567530854826784063836710, 8.267457766766554180261302096318
