| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 4·11-s + 14-s + 16-s − 6·17-s − 6·19-s − 4·22-s + 23-s − 5·25-s + 28-s − 10·29-s + 4·31-s + 32-s − 6·34-s − 2·37-s − 6·38-s + 10·41-s − 4·43-s − 4·44-s + 46-s − 12·47-s + 49-s − 5·50-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.20·11-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 1.37·19-s − 0.852·22-s + 0.208·23-s − 25-s + 0.188·28-s − 1.85·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.973·38-s + 1.56·41-s − 0.609·43-s − 0.603·44-s + 0.147·46-s − 1.75·47-s + 1/7·49-s − 0.707·50-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.234025981284700214770977709323, −7.64102126546899057073584795201, −6.75316689414728147191699147263, −6.02270300613256521695994463099, −5.20875944755922495345560120518, −4.48839957716031372476856482026, −3.74334454476541178742222592864, −2.51305341894706410491473493127, −1.92953756376400646693294624810, 0,
1.92953756376400646693294624810, 2.51305341894706410491473493127, 3.74334454476541178742222592864, 4.48839957716031372476856482026, 5.20875944755922495345560120518, 6.02270300613256521695994463099, 6.75316689414728147191699147263, 7.64102126546899057073584795201, 8.234025981284700214770977709323
