| L(s) = 1 | − 2-s + 4-s + 1.41·5-s − 8-s − 1.41·10-s − 11-s − 1.41·13-s + 16-s − 1.41·17-s + 2.82·19-s + 1.41·20-s + 22-s + 4·23-s − 2.99·25-s + 1.41·26-s + 6·29-s + 2.82·31-s − 32-s + 1.41·34-s − 8·37-s − 2.82·38-s − 1.41·40-s − 7.07·41-s − 8·43-s − 44-s − 4·46-s − 8.48·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.632·5-s − 0.353·8-s − 0.447·10-s − 0.301·11-s − 0.392·13-s + 0.250·16-s − 0.342·17-s + 0.648·19-s + 0.316·20-s + 0.213·22-s + 0.834·23-s − 0.599·25-s + 0.277·26-s + 1.11·29-s + 0.508·31-s − 0.176·32-s + 0.242·34-s − 1.31·37-s − 0.458·38-s − 0.223·40-s − 1.10·41-s − 1.21·43-s − 0.150·44-s − 0.589·46-s − 1.23·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 7.07T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 - 9.89T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.28221579696518488307160152922, −6.76647536949665559531313365175, −6.15855033748627495505040230747, −5.21118544326895154803339956925, −4.85308670455194117520888050716, −3.58566315054123885461219146000, −2.86985277158359714171875405724, −2.04319898960403537319275019497, −1.23854838287504997368745280975, 0,
1.23854838287504997368745280975, 2.04319898960403537319275019497, 2.86985277158359714171875405724, 3.58566315054123885461219146000, 4.85308670455194117520888050716, 5.21118544326895154803339956925, 6.15855033748627495505040230747, 6.76647536949665559531313365175, 7.28221579696518488307160152922
