L(s) = 1 | − 2.22·2-s − 3-s + 2.92·4-s − 0.165·5-s + 2.22·6-s − 2.06·8-s + 9-s + 0.368·10-s + 2.45·11-s − 2.92·12-s − 0.862·13-s + 0.165·15-s − 1.27·16-s − 4.02·17-s − 2.22·18-s − 2.72·19-s − 0.486·20-s − 5.43·22-s − 4.73·23-s + 2.06·24-s − 4.97·25-s + 1.91·26-s − 27-s − 6.41·29-s − 0.368·30-s − 4.79·31-s + 6.96·32-s + ⋯ |
L(s) = 1 | − 1.56·2-s − 0.577·3-s + 1.46·4-s − 0.0742·5-s + 0.906·6-s − 0.729·8-s + 0.333·9-s + 0.116·10-s + 0.738·11-s − 0.845·12-s − 0.239·13-s + 0.0428·15-s − 0.319·16-s − 0.975·17-s − 0.523·18-s − 0.625·19-s − 0.108·20-s − 1.15·22-s − 0.988·23-s + 0.421·24-s − 0.994·25-s + 0.375·26-s − 0.192·27-s − 1.19·29-s − 0.0672·30-s − 0.862·31-s + 1.23·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3086774315\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3086774315\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 + 2.22T + 2T^{2} \) |
| 5 | \( 1 + 0.165T + 5T^{2} \) |
| 11 | \( 1 - 2.45T + 11T^{2} \) |
| 13 | \( 1 + 0.862T + 13T^{2} \) |
| 17 | \( 1 + 4.02T + 17T^{2} \) |
| 19 | \( 1 + 2.72T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 + 6.41T + 29T^{2} \) |
| 31 | \( 1 + 4.79T + 31T^{2} \) |
| 37 | \( 1 + 7.78T + 37T^{2} \) |
| 43 | \( 1 - 7.56T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 6.13T + 53T^{2} \) |
| 59 | \( 1 + 8.52T + 59T^{2} \) |
| 61 | \( 1 - 0.773T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 4.99T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 5.29T + 83T^{2} \) |
| 89 | \( 1 + 9.60T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−8.145949427664635177263989042134, −7.48131718249941956598630682850, −6.81706666041392803083591203792, −6.26830555229923190210478010597, −5.41572943679356454307843934597, −4.36481439306826860958197789840, −3.69431339548009053371534647672, −2.16827599532079230008708779222, −1.70636355357256697438916742049, −0.37607161946350356870513374068,
0.37607161946350356870513374068, 1.70636355357256697438916742049, 2.16827599532079230008708779222, 3.69431339548009053371534647672, 4.36481439306826860958197789840, 5.41572943679356454307843934597, 6.26830555229923190210478010597, 6.81706666041392803083591203792, 7.48131718249941956598630682850, 8.145949427664635177263989042134