L(s) = 1 | + 2-s + 4-s − 3·5-s + 8-s − 3·10-s − 3·11-s + 5·13-s + 16-s + 3·17-s + 5·19-s − 3·20-s − 3·22-s − 3·23-s + 4·25-s + 5·26-s − 3·29-s − 4·31-s + 32-s + 3·34-s − 7·37-s + 5·38-s − 3·40-s − 9·41-s + 11·43-s − 3·44-s − 3·46-s + 4·50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.353·8-s − 0.948·10-s − 0.904·11-s + 1.38·13-s + 1/4·16-s + 0.727·17-s + 1.14·19-s − 0.670·20-s − 0.639·22-s − 0.625·23-s + 4/5·25-s + 0.980·26-s − 0.557·29-s − 0.718·31-s + 0.176·32-s + 0.514·34-s − 1.15·37-s + 0.811·38-s − 0.474·40-s − 1.40·41-s + 1.67·43-s − 0.452·44-s − 0.442·46-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.338313516\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.338313516\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.78029520578599438268505900096, −7.25732059508877738603477597734, −6.43473336818813678305531238320, −5.50918689656290729288515419866, −5.17380638337013071397783575342, −4.04707833211270402466104071972, −3.62966828052085670459801656385, −3.06325665649300207427668593005, −1.85381806023153311709754785900, −0.68096026423282212420123448045,
0.68096026423282212420123448045, 1.85381806023153311709754785900, 3.06325665649300207427668593005, 3.62966828052085670459801656385, 4.04707833211270402466104071972, 5.17380638337013071397783575342, 5.50918689656290729288515419866, 6.43473336818813678305531238320, 7.25732059508877738603477597734, 7.78029520578599438268505900096