| L(s) = 1 | + 0.242·2-s − 3-s − 1.94·4-s + 5-s − 0.242·6-s + 2.88·7-s − 0.955·8-s + 9-s + 0.242·10-s − 5.32·11-s + 1.94·12-s + 1.78·13-s + 0.700·14-s − 15-s + 3.65·16-s − 4.08·17-s + 0.242·18-s − 0.811·19-s − 1.94·20-s − 2.88·21-s − 1.29·22-s + 3.89·23-s + 0.955·24-s + 25-s + 0.432·26-s − 27-s − 5.60·28-s + ⋯ |
| L(s) = 1 | + 0.171·2-s − 0.577·3-s − 0.970·4-s + 0.447·5-s − 0.0989·6-s + 1.09·7-s − 0.337·8-s + 0.333·9-s + 0.0766·10-s − 1.60·11-s + 0.560·12-s + 0.494·13-s + 0.187·14-s − 0.258·15-s + 0.912·16-s − 0.991·17-s + 0.0571·18-s − 0.186·19-s − 0.434·20-s − 0.630·21-s − 0.275·22-s + 0.812·23-s + 0.195·24-s + 0.200·25-s + 0.0847·26-s − 0.192·27-s − 1.05·28-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 | \( 1 - 0.242T + 2T^{2} \) |
| 7 | \( 1 - 2.88T + 7T^{2} \) |
| 11 | \( 1 + 5.32T + 11T^{2} \) |
| 13 | \( 1 - 1.78T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 19 | \( 1 + 0.811T + 19T^{2} \) |
| 23 | \( 1 - 3.89T + 23T^{2} \) |
| 29 | \( 1 - 1.49T + 29T^{2} \) |
| 31 | \( 1 - 6.63T + 31T^{2} \) |
| 37 | \( 1 - 0.231T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 5.72T + 43T^{2} \) |
| 47 | \( 1 - 1.08T + 47T^{2} \) |
| 53 | \( 1 + 3.27T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 7.81T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 8.93T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 0.988T + 97T^{2} \) |
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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
−7.959069670261029558571102112400, −6.92056443558353032690238860745, −6.17362219583496250649011971070, −5.22618051478747091771722316285, −5.01167241986954545932524723380, −4.39801927781059907121121502190, −3.28148581992143978758098036728, −2.27385478221732681644145759883, −1.20440243999914163765101945443, 0,
1.20440243999914163765101945443, 2.27385478221732681644145759883, 3.28148581992143978758098036728, 4.39801927781059907121121502190, 5.01167241986954545932524723380, 5.22618051478747091771722316285, 6.17362219583496250649011971070, 6.92056443558353032690238860745, 7.959069670261029558571102112400
