| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s + 4.82·11-s − 4.82·13-s − 15-s − 2.82·17-s − 2·21-s + 5.65·23-s + 25-s + 27-s − 7.65·29-s − 6.82·31-s + 4.82·33-s + 2·35-s + 10.4·37-s − 4.82·39-s + 7.65·41-s − 9.65·43-s − 45-s − 9.65·47-s − 3·49-s − 2.82·51-s − 0.343·53-s − 4.82·55-s − 0.828·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 0.333·9-s + 1.45·11-s − 1.33·13-s − 0.258·15-s − 0.685·17-s − 0.436·21-s + 1.17·23-s + 0.200·25-s + 0.192·27-s − 1.42·29-s − 1.22·31-s + 0.840·33-s + 0.338·35-s + 1.72·37-s − 0.773·39-s + 1.19·41-s − 1.47·43-s − 0.149·45-s − 1.40·47-s − 0.428·49-s − 0.396·51-s − 0.0471·53-s − 0.651·55-s − 0.107·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 + 9.65T + 47T^{2} \) |
| 53 | \( 1 + 0.343T + 53T^{2} \) |
| 59 | \( 1 + 0.828T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 + 2.34T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 4.48T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−8.105500556012422632035757697208, −7.26267370186674390213852162645, −6.87746432065015431590065997132, −6.04134107516515135099198813769, −4.92632358264519517447241518087, −4.16593155706803711257900199501, −3.43448622087503449119383155452, −2.62281502463820647787683475795, −1.50970162287548745186148872109, 0,
1.50970162287548745186148872109, 2.62281502463820647787683475795, 3.43448622087503449119383155452, 4.16593155706803711257900199501, 4.92632358264519517447241518087, 6.04134107516515135099198813769, 6.87746432065015431590065997132, 7.26267370186674390213852162645, 8.105500556012422632035757697208
