| L(s) = 1 | − 2·2-s − 1.20·3-s + 4·4-s + 2.40·6-s − 8·8-s − 25.5·9-s − 26.4·11-s − 4.80·12-s − 55.0·13-s + 16·16-s − 49.4·17-s + 51.1·18-s − 5.46·19-s + 52.8·22-s + 1.07·23-s + 9.60·24-s + 110.·26-s + 63.0·27-s − 246.·29-s − 228.·31-s − 32·32-s + 31.7·33-s + 98.8·34-s − 102.·36-s − 381.·37-s + 10.9·38-s + 66.0·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.230·3-s + 0.5·4-s + 0.163·6-s − 0.353·8-s − 0.946·9-s − 0.724·11-s − 0.115·12-s − 1.17·13-s + 0.250·16-s − 0.705·17-s + 0.669·18-s − 0.0659·19-s + 0.512·22-s + 0.00972·23-s + 0.0816·24-s + 0.830·26-s + 0.449·27-s − 1.57·29-s − 1.32·31-s − 0.176·32-s + 0.167·33-s + 0.498·34-s − 0.473·36-s − 1.69·37-s + 0.0466·38-s + 0.271·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1024290265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1024290265\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.20T + 27T^{2} \) |
| 11 | \( 1 + 26.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 55.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 49.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.46T + 6.85e3T^{2} \) |
| 23 | \( 1 - 1.07T + 1.21e4T^{2} \) |
| 29 | \( 1 + 246.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 228.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 381.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 48.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 233.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 245.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 269.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 784.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 425.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 918.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 200.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 778.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 189.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 611.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 292.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.42e3T + 9.12e5T^{2} \) |
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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
−8.798984272190429633499930517528, −7.68533170619529012989195755386, −7.35293320149277509870245209892, −6.31330559650646712409316896106, −5.50951006470474822831430069590, −4.85442783535994326113390501513, −3.54206651933313801797867879338, −2.58389977868574880594252380300, −1.80295437021985133133198777647, −0.14894607776031489325557461598,
0.14894607776031489325557461598, 1.80295437021985133133198777647, 2.58389977868574880594252380300, 3.54206651933313801797867879338, 4.85442783535994326113390501513, 5.50951006470474822831430069590, 6.31330559650646712409316896106, 7.35293320149277509870245209892, 7.68533170619529012989195755386, 8.798984272190429633499930517528
