L(s) = 1 | − 1.77·2-s − 3-s + 1.14·4-s + 0.589·5-s + 1.77·6-s + 7-s + 1.52·8-s + 9-s − 1.04·10-s − 1.14·12-s + 1.54·13-s − 1.77·14-s − 0.589·15-s − 4.98·16-s − 7.14·17-s − 1.77·18-s + 6.19·19-s + 0.672·20-s − 21-s + 3.76·23-s − 1.52·24-s − 4.65·25-s − 2.73·26-s − 27-s + 1.14·28-s − 0.607·29-s + 1.04·30-s + ⋯ |
L(s) = 1 | − 1.25·2-s − 0.577·3-s + 0.570·4-s + 0.263·5-s + 0.723·6-s + 0.377·7-s + 0.538·8-s + 0.333·9-s − 0.330·10-s − 0.329·12-s + 0.428·13-s − 0.473·14-s − 0.152·15-s − 1.24·16-s − 1.73·17-s − 0.417·18-s + 1.42·19-s + 0.150·20-s − 0.218·21-s + 0.784·23-s − 0.310·24-s − 0.930·25-s − 0.536·26-s − 0.192·27-s + 0.215·28-s − 0.112·29-s + 0.190·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 5 | \( 1 - 0.589T + 5T^{2} \) |
| 13 | \( 1 - 1.54T + 13T^{2} \) |
| 17 | \( 1 + 7.14T + 17T^{2} \) |
| 19 | \( 1 - 6.19T + 19T^{2} \) |
| 23 | \( 1 - 3.76T + 23T^{2} \) |
| 29 | \( 1 + 0.607T + 29T^{2} \) |
| 31 | \( 1 + 6.87T + 31T^{2} \) |
| 37 | \( 1 + 7.70T + 37T^{2} \) |
| 41 | \( 1 + 7.48T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 7.04T + 53T^{2} \) |
| 59 | \( 1 + 3.35T + 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 + 2.19T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 2.58T + 73T^{2} \) |
| 79 | \( 1 - 4.66T + 79T^{2} \) |
| 83 | \( 1 - 3.50T + 83T^{2} \) |
| 89 | \( 1 - 8.15T + 89T^{2} \) |
| 97 | \( 1 + 4.79T + 97T^{2} \) |
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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
−8.895191020123332503171736964800, −7.74697522532409763567821138504, −7.20214952084966291160660733195, −6.43133209355336179953748692762, −5.40136048818213573354215019724, −4.71460267500215032479234558573, −3.65332138961462775921508814073, −2.14876146160426887300533742595, −1.29768083078688157214873312145, 0,
1.29768083078688157214873312145, 2.14876146160426887300533742595, 3.65332138961462775921508814073, 4.71460267500215032479234558573, 5.40136048818213573354215019724, 6.43133209355336179953748692762, 7.20214952084966291160660733195, 7.74697522532409763567821138504, 8.895191020123332503171736964800