| L(s) = 1 | + 2-s − 3-s + 4-s − 1.49·5-s − 6-s + 8-s + 9-s − 1.49·10-s + 5.90·11-s − 12-s − 0.505·13-s + 1.49·15-s + 16-s − 6.55·17-s + 18-s − 2·19-s − 1.49·20-s + 5.90·22-s − 23-s − 24-s − 2.76·25-s − 0.505·26-s − 27-s + 7.03·29-s + 1.49·30-s − 5.26·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.668·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.472·10-s + 1.78·11-s − 0.288·12-s − 0.140·13-s + 0.385·15-s + 0.250·16-s − 1.58·17-s + 0.235·18-s − 0.458·19-s − 0.334·20-s + 1.25·22-s − 0.208·23-s − 0.204·24-s − 0.553·25-s − 0.0992·26-s − 0.192·27-s + 1.30·29-s + 0.272·30-s − 0.945·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 1.49T + 5T^{2} \) |
| 11 | \( 1 - 5.90T + 11T^{2} \) |
| 13 | \( 1 + 0.505T + 13T^{2} \) |
| 17 | \( 1 + 6.55T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 - 7.03T + 29T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 + 9.95T + 37T^{2} \) |
| 41 | \( 1 - 3.05T + 41T^{2} \) |
| 43 | \( 1 + 8.96T + 43T^{2} \) |
| 47 | \( 1 - 7.30T + 47T^{2} \) |
| 53 | \( 1 - 4.91T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 0.458T + 61T^{2} \) |
| 67 | \( 1 + 6.18T + 67T^{2} \) |
| 71 | \( 1 + 9.10T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 19.1T + 97T^{2} \) |
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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
−7.17648853053727034501237473820, −6.90167084113978340443943711269, −6.26494607777064839359160874416, −5.50874753543384454169406578765, −4.53449067684653035382903269130, −4.13531101339035048897633406660, −3.51569374475604985635268536291, −2.28567286846969769122911738192, −1.37831191702156250099468176372, 0,
1.37831191702156250099468176372, 2.28567286846969769122911738192, 3.51569374475604985635268536291, 4.13531101339035048897633406660, 4.53449067684653035382903269130, 5.50874753543384454169406578765, 6.26494607777064839359160874416, 6.90167084113978340443943711269, 7.17648853053727034501237473820
