| L(s) = 1 | + 2.62e5·2-s − 7.64e8·3-s + 6.87e10·4-s − 2.00e14·6-s + 1.80e16·8-s + 4.33e17·9-s − 1.09e19·11-s − 5.25e19·12-s + 4.72e21·16-s − 1.37e22·17-s + 1.13e23·18-s − 9.60e22·19-s − 2.88e24·22-s − 1.37e25·24-s + 1.45e25·25-s − 2.16e26·27-s + 1.23e27·32-s + 8.40e27·33-s − 3.60e27·34-s + 2.98e28·36-s − 2.51e28·38-s + 5.82e28·41-s + 4.95e29·43-s − 7.55e29·44-s − 3.60e30·48-s + 2.65e30·49-s + 3.81e30·50-s + ⋯ |
| L(s) = 1 | + 2-s − 1.97·3-s + 4-s − 1.97·6-s + 8-s + 2.89·9-s − 1.97·11-s − 1.97·12-s + 16-s − 0.978·17-s + 2.89·18-s − 0.922·19-s − 1.97·22-s − 1.97·24-s + 25-s − 3.72·27-s + 32-s + 3.90·33-s − 0.978·34-s + 2.89·36-s − 0.922·38-s + 0.543·41-s + 1.96·43-s − 1.97·44-s − 1.97·48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{37}{2})\) |
\(\approx\) |
\(1.569553246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.569553246\) |
| \(L(19)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{18} T \) |
| good | 3 | \( 1 + 764174222 T + p^{36} T^{2} \) |
| 5 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 7 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 11 | \( 1 + 10994270975207717038 T + p^{36} T^{2} \) |
| 13 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 17 | \( 1 + \)\(13\!\cdots\!82\)\( T + p^{36} T^{2} \) |
| 19 | \( 1 + \)\(96\!\cdots\!18\)\( T + p^{36} T^{2} \) |
| 23 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 29 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 31 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 37 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 41 | \( 1 - \)\(58\!\cdots\!42\)\( T + p^{36} T^{2} \) |
| 43 | \( 1 - \)\(49\!\cdots\!98\)\( T + p^{36} T^{2} \) |
| 47 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 53 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 59 | \( 1 + \)\(45\!\cdots\!58\)\( T + p^{36} T^{2} \) |
| 61 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 67 | \( 1 - \)\(10\!\cdots\!18\)\( T + p^{36} T^{2} \) |
| 71 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 73 | \( 1 + \)\(32\!\cdots\!62\)\( T + p^{36} T^{2} \) |
| 79 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 83 | \( 1 - \)\(54\!\cdots\!18\)\( T + p^{36} T^{2} \) |
| 89 | \( 1 + \)\(82\!\cdots\!38\)\( T + p^{36} T^{2} \) |
| 97 | \( 1 - \)\(10\!\cdots\!78\)\( T + p^{36} T^{2} \) |
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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
−13.15146126973858412876325122418, −12.47183012763049727824026172483, −11.00726212639832365631634964578, −10.51009276959672229734315989493, −7.40315005537589028508819768215, −6.21853825108440747147589484958, −5.23119826766625364033267824155, −4.37802236870706781924294801668, −2.28399701156989347761337517617, −0.61207161456931348682760341054,
0.61207161456931348682760341054, 2.28399701156989347761337517617, 4.37802236870706781924294801668, 5.23119826766625364033267824155, 6.21853825108440747147589484958, 7.40315005537589028508819768215, 10.51009276959672229734315989493, 11.00726212639832365631634964578, 12.47183012763049727824026172483, 13.15146126973858412876325122418
