| L(s) = 1 | + 2-s + 3-s + 4-s − 1.91·5-s + 6-s − 1.73·7-s + 8-s + 9-s − 1.91·10-s + 1.28·11-s + 12-s + 5.11·13-s − 1.73·14-s − 1.91·15-s + 16-s − 17-s + 18-s − 7.73·19-s − 1.91·20-s − 1.73·21-s + 1.28·22-s − 6.84·23-s + 24-s − 1.32·25-s + 5.11·26-s + 27-s − 1.73·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.857·5-s + 0.408·6-s − 0.656·7-s + 0.353·8-s + 0.333·9-s − 0.606·10-s + 0.387·11-s + 0.288·12-s + 1.41·13-s − 0.464·14-s − 0.495·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 1.77·19-s − 0.428·20-s − 0.379·21-s + 0.273·22-s − 1.42·23-s + 0.204·24-s − 0.264·25-s + 1.00·26-s + 0.192·27-s − 0.328·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 + 1.91T + 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 - 1.28T + 11T^{2} \) |
| 13 | \( 1 - 5.11T + 13T^{2} \) |
| 19 | \( 1 + 7.73T + 19T^{2} \) |
| 23 | \( 1 + 6.84T + 23T^{2} \) |
| 29 | \( 1 + 8.97T + 29T^{2} \) |
| 31 | \( 1 + 0.566T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 1.72T + 41T^{2} \) |
| 43 | \( 1 - 7.67T + 43T^{2} \) |
| 47 | \( 1 + 3.75T + 47T^{2} \) |
| 53 | \( 1 + 1.44T + 53T^{2} \) |
| 61 | \( 1 - 3.44T + 61T^{2} \) |
| 67 | \( 1 + 1.48T + 67T^{2} \) |
| 71 | \( 1 + 5.61T + 71T^{2} \) |
| 73 | \( 1 + 7.84T + 73T^{2} \) |
| 79 | \( 1 + 1.87T + 79T^{2} \) |
| 83 | \( 1 - 1.97T + 83T^{2} \) |
| 89 | \( 1 + 2.34T + 89T^{2} \) |
| 97 | \( 1 + 7.86T + 97T^{2} \) |
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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
−7.83641471356857613654581457190, −6.92237244982155578934184990069, −6.19755597752462889346972760281, −5.82195488511012258365906322810, −4.35702216676201136464338814674, −4.01595987136681055146961382116, −3.52265342412230438634782543711, −2.49517112455505472614658020780, −1.58777900756675827738509875206, 0,
1.58777900756675827738509875206, 2.49517112455505472614658020780, 3.52265342412230438634782543711, 4.01595987136681055146961382116, 4.35702216676201136464338814674, 5.82195488511012258365906322810, 6.19755597752462889346972760281, 6.92237244982155578934184990069, 7.83641471356857613654581457190
