| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 12-s − 15-s + 16-s + 18-s − 19-s + 20-s − 24-s + 25-s − 27-s − 30-s + 32-s + 36-s − 38-s + 40-s + 45-s − 48-s − 49-s + 50-s − 2·53-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 12-s − 15-s + 16-s + 18-s − 19-s + 20-s − 24-s + 25-s − 27-s − 30-s + 32-s + 36-s − 38-s + 40-s + 45-s − 48-s − 49-s + 50-s − 2·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.018042458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.018042458\) |
| \(L(1)\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 + T )^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 - T )^{2} \) |
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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
−9.507478946686207961565764586678, −8.295731784986999892996448600951, −7.28300260395652118511101627984, −6.42657800462644993038129410790, −6.14722947391243799699289088529, −5.18964578902177367574647314192, −4.68567324389578924997117958939, −3.64302377183107483406632685597, −2.39560782030209754125253618298, −1.46105579814172982662139019744,
1.46105579814172982662139019744, 2.39560782030209754125253618298, 3.64302377183107483406632685597, 4.68567324389578924997117958939, 5.18964578902177367574647314192, 6.14722947391243799699289088529, 6.42657800462644993038129410790, 7.28300260395652118511101627984, 8.295731784986999892996448600951, 9.507478946686207961565764586678
