| L(s) = 1 | + 2.74·3-s + 4.54·9-s − 7.03·13-s − 5·25-s + 4.23·27-s − 3.94·29-s − 5.83·31-s − 19.3·39-s + 12.5·41-s + 0.340·47-s − 7·49-s − 12·59-s − 16.8·71-s + 9.44·73-s − 13.7·75-s − 1.98·81-s − 10.8·87-s − 16.0·93-s + 6·101-s − 31.9·117-s + ⋯ |
| L(s) = 1 | + 1.58·3-s + 1.51·9-s − 1.95·13-s − 25-s + 0.815·27-s − 0.733·29-s − 1.04·31-s − 3.09·39-s + 1.95·41-s + 0.0496·47-s − 49-s − 1.56·59-s − 1.99·71-s + 1.10·73-s − 1.58·75-s − 0.220·81-s − 1.16·87-s − 1.66·93-s + 0.597·101-s − 2.95·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2.74T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 7.03T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 + 3.94T + 29T^{2} \) |
| 31 | \( 1 + 5.83T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 0.340T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 16.8T + 71T^{2} \) |
| 73 | \( 1 - 9.44T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.67085774603469173661777256258, −7.14149395669653914525891601012, −6.11698302638617117510216234269, −5.24001328653230485534456117153, −4.44043820556751834371345187469, −3.78437663528822573077788856126, −2.93403621453518468232177598011, −2.34939753285961251036450481855, −1.65376265923042609926210004975, 0,
1.65376265923042609926210004975, 2.34939753285961251036450481855, 2.93403621453518468232177598011, 3.78437663528822573077788856126, 4.44043820556751834371345187469, 5.24001328653230485534456117153, 6.11698302638617117510216234269, 7.14149395669653914525891601012, 7.67085774603469173661777256258
