| L(s) = 1 | − 7·2-s + 9·3-s + 33·4-s + 25·5-s − 63·6-s − 119·8-s + 81·9-s − 175·10-s + 297·12-s + 225·15-s + 305·16-s − 382·17-s − 567·18-s − 238·19-s + 825·20-s + 98·23-s − 1.07e3·24-s + 625·25-s + 729·27-s − 1.57e3·30-s − 1.91e3·31-s − 231·32-s + 2.67e3·34-s + 2.67e3·36-s + 1.66e3·38-s − 2.97e3·40-s + 2.02e3·45-s + ⋯ |
| L(s) = 1 | − 7/4·2-s + 3-s + 2.06·4-s + 5-s − 7/4·6-s − 1.85·8-s + 9-s − 7/4·10-s + 2.06·12-s + 15-s + 1.19·16-s − 1.32·17-s − 7/4·18-s − 0.659·19-s + 2.06·20-s + 0.185·23-s − 1.85·24-s + 25-s + 27-s − 7/4·30-s − 1.99·31-s − 0.225·32-s + 2.31·34-s + 2.06·36-s + 1.15·38-s − 1.85·40-s + 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8253872992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8253872992\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 + 7 T + p^{4} T^{2} \) |
| 7 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( 1 + 382 T + p^{4} T^{2} \) |
| 19 | \( 1 + 238 T + p^{4} T^{2} \) |
| 23 | \( 1 - 98 T + p^{4} T^{2} \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( 1 + 1918 T + p^{4} T^{2} \) |
| 37 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( 1 + 4222 T + p^{4} T^{2} \) |
| 53 | \( 1 - 1778 T + p^{4} T^{2} \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 - 6482 T + p^{4} T^{2} \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 79 | \( 1 + 2878 T + p^{4} T^{2} \) |
| 83 | \( 1 - 9938 T + p^{4} T^{2} \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
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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
−18.51634148406880292609875452827, −17.62660389214601930903577643589, −16.30297059041558398067348571732, −14.85685512633365025684996031893, −13.16566330778350995554098775537, −10.76777679784508316400606711237, −9.510102592243532002975505857194, −8.585971616843123469031927456283, −6.89448545726658441111330808012, −2.06476355225246860341332385126,
2.06476355225246860341332385126, 6.89448545726658441111330808012, 8.585971616843123469031927456283, 9.510102592243532002975505857194, 10.76777679784508316400606711237, 13.16566330778350995554098775537, 14.85685512633365025684996031893, 16.30297059041558398067348571732, 17.62660389214601930903577643589, 18.51634148406880292609875452827
