| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s − 4·11-s + 6·13-s − 2·14-s + 16-s + 4·17-s + 19-s + 4·22-s + 4·23-s − 6·26-s + 2·28-s − 6·29-s − 6·31-s − 32-s − 4·34-s − 10·37-s − 38-s − 4·41-s − 12·43-s − 4·44-s − 4·46-s + 4·47-s − 3·49-s + 6·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 1.20·11-s + 1.66·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s + 0.229·19-s + 0.852·22-s + 0.834·23-s − 1.17·26-s + 0.377·28-s − 1.11·29-s − 1.07·31-s − 0.176·32-s − 0.685·34-s − 1.64·37-s − 0.162·38-s − 0.624·41-s − 1.82·43-s − 0.603·44-s − 0.589·46-s + 0.583·47-s − 3/7·49-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.69787464173758314207740152450, −6.92465975204620545886021919982, −6.06308608778305379850346973105, −5.37817775904103702307919769295, −4.86802437783647759141855172545, −3.53523660403903799144368606175, −3.19145578399769463338290811972, −1.85387130165108636351727613914, −1.35376053111283218101424788308, 0,
1.35376053111283218101424788308, 1.85387130165108636351727613914, 3.19145578399769463338290811972, 3.53523660403903799144368606175, 4.86802437783647759141855172545, 5.37817775904103702307919769295, 6.06308608778305379850346973105, 6.92465975204620545886021919982, 7.69787464173758314207740152450
