| L(s) = 1 | + 0.806·2-s − 3-s − 1.34·4-s + 5-s − 0.806·6-s + 2.45·7-s − 2.70·8-s + 9-s + 0.806·10-s + 2.93·11-s + 1.34·12-s − 1.89·13-s + 1.97·14-s − 15-s + 0.519·16-s − 1.43·17-s + 0.806·18-s − 2.97·19-s − 1.34·20-s − 2.45·21-s + 2.36·22-s − 6.45·23-s + 2.70·24-s + 25-s − 1.52·26-s − 27-s − 3.30·28-s + ⋯ |
| L(s) = 1 | + 0.570·2-s − 0.577·3-s − 0.674·4-s + 0.447·5-s − 0.329·6-s + 0.926·7-s − 0.955·8-s + 0.333·9-s + 0.255·10-s + 0.884·11-s + 0.389·12-s − 0.524·13-s + 0.528·14-s − 0.258·15-s + 0.129·16-s − 0.347·17-s + 0.190·18-s − 0.682·19-s − 0.301·20-s − 0.535·21-s + 0.504·22-s − 1.34·23-s + 0.551·24-s + 0.200·25-s − 0.299·26-s − 0.192·27-s − 0.625·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 0.806T + 2T^{2} \) |
| 7 | \( 1 - 2.45T + 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 13 | \( 1 + 1.89T + 13T^{2} \) |
| 17 | \( 1 + 1.43T + 17T^{2} \) |
| 19 | \( 1 + 2.97T + 19T^{2} \) |
| 23 | \( 1 + 6.45T + 23T^{2} \) |
| 29 | \( 1 + 1.21T + 29T^{2} \) |
| 31 | \( 1 - 3.43T + 31T^{2} \) |
| 37 | \( 1 + 0.0402T + 37T^{2} \) |
| 41 | \( 1 + 2.33T + 41T^{2} \) |
| 43 | \( 1 - 9.77T + 43T^{2} \) |
| 47 | \( 1 + 8.78T + 47T^{2} \) |
| 53 | \( 1 + 2.31T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 + 3.11T + 61T^{2} \) |
| 67 | \( 1 - 0.311T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 2.72T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 6.48T + 83T^{2} \) |
| 89 | \( 1 + 3.96T + 89T^{2} \) |
| 97 | \( 1 - 6.61T + 97T^{2} \) |
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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
−7.81420421701201960495422788519, −6.74406367947499770244014032987, −6.12536938686258589399649362462, −5.56668862113245604712298748633, −4.54014197919663176490014086263, −4.51600144003191066172210888265, −3.46553054764496602835326459083, −2.26305799464759935115011574462, −1.35199118711217528080828756259, 0,
1.35199118711217528080828756259, 2.26305799464759935115011574462, 3.46553054764496602835326459083, 4.51600144003191066172210888265, 4.54014197919663176490014086263, 5.56668862113245604712298748633, 6.12536938686258589399649362462, 6.74406367947499770244014032987, 7.81420421701201960495422788519
