| L(s) = 1 | + 1.41·2-s − 3-s − 5-s − 1.41·6-s − 2.82·8-s + 9-s − 1.41·10-s + 3.41·11-s − 1.58·13-s + 15-s − 4.00·16-s − 6.24·17-s + 1.41·18-s − 6.65·19-s + 4.82·22-s − 6.24·23-s + 2.82·24-s + 25-s − 2.24·26-s − 27-s − 0.242·29-s + 1.41·30-s − 0.171·31-s − 3.41·33-s − 8.82·34-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 0.577·3-s − 0.447·5-s − 0.577·6-s − 0.999·8-s + 0.333·9-s − 0.447·10-s + 1.02·11-s − 0.439·13-s + 0.258·15-s − 1.00·16-s − 1.51·17-s + 0.333·18-s − 1.52·19-s + 1.02·22-s − 1.30·23-s + 0.577·24-s + 0.200·25-s − 0.439·26-s − 0.192·27-s − 0.0450·29-s + 0.258·30-s − 0.0308·31-s − 0.594·33-s − 1.51·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 + 6.65T + 19T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 + 0.242T + 29T^{2} \) |
| 31 | \( 1 + 0.171T + 31T^{2} \) |
| 37 | \( 1 + 5.58T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 9.31T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 3.41T + 71T^{2} \) |
| 73 | \( 1 - 2.07T + 73T^{2} \) |
| 79 | \( 1 + 4.65T + 79T^{2} \) |
| 83 | \( 1 - 5.41T + 83T^{2} \) |
| 89 | \( 1 + 3.75T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−10.07996709646415843023796327995, −9.027407153440427529431260797103, −8.340158390294596929056689449612, −6.84251427864877263877490124754, −6.38649506537361641389162733226, −5.30010439325129643550210623637, −4.28365844069808670777336448311, −3.88443567331223310588735779556, −2.22463426679336255486529969906, 0,
2.22463426679336255486529969906, 3.88443567331223310588735779556, 4.28365844069808670777336448311, 5.30010439325129643550210623637, 6.38649506537361641389162733226, 6.84251427864877263877490124754, 8.340158390294596929056689449612, 9.027407153440427529431260797103, 10.07996709646415843023796327995
