| L(s) = 1 | − 3-s + 5-s + 4.46·7-s + 9-s − 6.19·11-s − 6.73·13-s − 15-s − 4.26·17-s − 2.73·19-s − 4.46·21-s − 23-s + 25-s − 27-s − 3.19·29-s − 31-s + 6.19·33-s + 4.46·35-s + 9.39·37-s + 6.73·39-s − 4.26·41-s − 3.46·43-s + 45-s − 2.73·47-s + 12.9·49-s + 4.26·51-s − 10.6·53-s − 6.19·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.68·7-s + 0.333·9-s − 1.86·11-s − 1.86·13-s − 0.258·15-s − 1.03·17-s − 0.626·19-s − 0.974·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s − 0.593·29-s − 0.179·31-s + 1.07·33-s + 0.754·35-s + 1.54·37-s + 1.07·39-s − 0.666·41-s − 0.528·43-s + 0.149·45-s − 0.398·47-s + 1.84·49-s + 0.597·51-s − 1.46·53-s − 0.835·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 4.46T + 7T^{2} \) |
| 11 | \( 1 + 6.19T + 11T^{2} \) |
| 13 | \( 1 + 6.73T + 13T^{2} \) |
| 17 | \( 1 + 4.26T + 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 29 | \( 1 + 3.19T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 - 9.39T + 37T^{2} \) |
| 41 | \( 1 + 4.26T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 + 2.73T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 3.19T + 59T^{2} \) |
| 61 | \( 1 + 7.26T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 4.39T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−9.277365480394601623498810940592, −8.051874359199637072064102805641, −7.76380641210151861002896416895, −6.76450660636396141453867095605, −5.57297162566067886575550553229, −4.93860962938114778536749773258, −4.52290958428577863874563109235, −2.56398691011174799084950819833, −1.91439159429413950983027089324, 0,
1.91439159429413950983027089324, 2.56398691011174799084950819833, 4.52290958428577863874563109235, 4.93860962938114778536749773258, 5.57297162566067886575550553229, 6.76450660636396141453867095605, 7.76380641210151861002896416895, 8.051874359199637072064102805641, 9.277365480394601623498810940592
