| L(s) = 1 | − 2.39e5·2-s − 1.29e8·3-s + 4.02e10·4-s + 7.62e11·5-s + 3.09e13·6-s − 5.53e15·8-s + 1.66e16·9-s − 1.82e17·10-s − 5.20e18·12-s − 9.85e19·15-s + 6.35e20·16-s − 1.32e21·17-s − 3.99e21·18-s + 9.62e21·19-s + 3.07e22·20-s + 2.82e23·23-s + 7.14e23·24-s + 5.82e23·25-s − 2.15e24·27-s + 2.36e25·30-s + 2.35e25·31-s − 5.71e25·32-s + 3.18e26·34-s + 6.71e26·36-s − 2.30e27·38-s − 4.22e27·40-s + 1.27e28·45-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 3-s + 2.34·4-s + 5-s + 1.82·6-s − 2.45·8-s + 9-s − 1.82·10-s − 2.34·12-s − 15-s + 2.15·16-s − 1.60·17-s − 1.82·18-s + 1.75·19-s + 2.34·20-s + 1.99·23-s + 2.45·24-s + 25-s − 27-s + 1.82·30-s + 1.04·31-s − 1.47·32-s + 2.93·34-s + 2.34·36-s − 3.21·38-s − 2.45·40-s + 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{35}{2})\) |
\(\approx\) |
\(0.8556020332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8556020332\) |
| \(L(18)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{17} T \) |
| 5 | \( 1 - p^{17} T \) |
| good | 2 | \( 1 + 239699 T + p^{34} T^{2} \) |
| 7 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 11 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 13 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 17 | \( 1 + \)\(13\!\cdots\!74\)\( T + p^{34} T^{2} \) |
| 19 | \( 1 - \)\(96\!\cdots\!38\)\( T + p^{34} T^{2} \) |
| 23 | \( 1 - \)\(28\!\cdots\!14\)\( T + p^{34} T^{2} \) |
| 29 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 31 | \( 1 - \)\(23\!\cdots\!82\)\( T + p^{34} T^{2} \) |
| 37 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 41 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 43 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 47 | \( 1 + \)\(30\!\cdots\!54\)\( T + p^{34} T^{2} \) |
| 53 | \( 1 - \)\(15\!\cdots\!54\)\( T + p^{34} T^{2} \) |
| 59 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 61 | \( 1 - \)\(15\!\cdots\!82\)\( T + p^{34} T^{2} \) |
| 67 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 71 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 73 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 79 | \( 1 + \)\(33\!\cdots\!22\)\( T + p^{34} T^{2} \) |
| 83 | \( 1 - \)\(82\!\cdots\!34\)\( T + p^{34} T^{2} \) |
| 89 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 97 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
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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
−11.64686000455722507481186805473, −10.76640517147164735847743716745, −9.773164983836162761019795738811, −8.896840566290807859914145045198, −7.21290204286484371027208377174, −6.44752394089391359510538076453, −5.11966785379544511903249799154, −2.64493295195558494984177194021, −1.40585396394185426683415418766, −0.67480363475196286399791994967,
0.67480363475196286399791994967, 1.40585396394185426683415418766, 2.64493295195558494984177194021, 5.11966785379544511903249799154, 6.44752394089391359510538076453, 7.21290204286484371027208377174, 8.896840566290807859914145045198, 9.773164983836162761019795738811, 10.76640517147164735847743716745, 11.64686000455722507481186805473
