| L(s) = 1 | − 0.460·2-s − 0.539·3-s − 1.78·4-s − 4.10·5-s + 0.248·6-s + 1.74·8-s − 2.70·9-s + 1.88·10-s + 2.76·11-s + 0.965·12-s + 4.05·13-s + 2.21·15-s + 2.77·16-s − 5.22·17-s + 1.24·18-s + 0.109·19-s + 7.33·20-s − 1.27·22-s + 6.08·23-s − 0.941·24-s + 11.8·25-s − 1.86·26-s + 3.08·27-s + 2.25·29-s − 1.01·30-s + 1.18·31-s − 4.76·32-s + ⋯ |
| L(s) = 1 | − 0.325·2-s − 0.311·3-s − 0.894·4-s − 1.83·5-s + 0.101·6-s + 0.616·8-s − 0.902·9-s + 0.597·10-s + 0.834·11-s + 0.278·12-s + 1.12·13-s + 0.571·15-s + 0.693·16-s − 1.26·17-s + 0.293·18-s + 0.0250·19-s + 1.63·20-s − 0.271·22-s + 1.26·23-s − 0.192·24-s + 2.36·25-s − 0.366·26-s + 0.592·27-s + 0.418·29-s − 0.186·30-s + 0.212·31-s − 0.842·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 0.460T + 2T^{2} \) |
| 3 | \( 1 + 0.539T + 3T^{2} \) |
| 5 | \( 1 + 4.10T + 5T^{2} \) |
| 11 | \( 1 - 2.76T + 11T^{2} \) |
| 13 | \( 1 - 4.05T + 13T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 - 0.109T + 19T^{2} \) |
| 23 | \( 1 - 6.08T + 23T^{2} \) |
| 29 | \( 1 - 2.25T + 29T^{2} \) |
| 31 | \( 1 - 1.18T + 31T^{2} \) |
| 37 | \( 1 + 8.95T + 37T^{2} \) |
| 43 | \( 1 + 7.93T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 6.23T + 53T^{2} \) |
| 59 | \( 1 - 9.43T + 59T^{2} \) |
| 61 | \( 1 + 6.89T + 61T^{2} \) |
| 67 | \( 1 + 1.35T + 67T^{2} \) |
| 71 | \( 1 + 7.90T + 71T^{2} \) |
| 73 | \( 1 + 9.88T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 0.852T + 89T^{2} \) |
| 97 | \( 1 + 9.92T + 97T^{2} \) |
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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
−8.695935683602694694538501102734, −8.325020777682234876030365011986, −7.25037890503294240867123163741, −6.59726751178773381521546329118, −5.40908198310232626954349656470, −4.49966951751074924821877384455, −3.90201972871781149247036894347, −3.10596952704629512806139718781, −1.06648634595800888515613576231, 0,
1.06648634595800888515613576231, 3.10596952704629512806139718781, 3.90201972871781149247036894347, 4.49966951751074924821877384455, 5.40908198310232626954349656470, 6.59726751178773381521546329118, 7.25037890503294240867123163741, 8.325020777682234876030365011986, 8.695935683602694694538501102734
