| L(s) = 1 | + 3.97e13·3-s + 2.88e17·4-s − 8.73e19·5-s − 3.12e27·9-s − 1.58e30·11-s + 1.14e31·12-s − 3.47e33·15-s + 8.30e34·16-s − 2.51e37·20-s − 3.57e38·23-s − 2.70e40·25-s − 3.11e41·27-s + 1.06e43·31-s − 6.30e43·33-s − 9.01e44·36-s + 3.23e45·37-s − 4.57e47·44-s + 2.73e47·45-s − 3.27e48·47-s + 3.30e48·48-s + 1.03e49·49-s − 1.77e50·53-s + 1.38e50·55-s + 4.51e51·59-s − 1.00e51·60-s + 2.39e52·64-s + 1.77e53·67-s + ⋯ |
| L(s) = 1 | + 0.579·3-s + 4-s − 0.469·5-s − 0.664·9-s − 11-s + 0.579·12-s − 0.271·15-s + 16-s − 0.469·20-s − 0.115·23-s − 0.779·25-s − 0.964·27-s + 0.600·31-s − 0.579·33-s − 0.664·36-s + 1.07·37-s − 44-s + 0.311·45-s − 1.05·47-s + 0.579·48-s + 49-s − 1.75·53-s + 0.469·55-s + 1.99·59-s − 0.271·60-s + 64-s + 1.96·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(59-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+29) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{59}{2})\) |
\(\approx\) |
\(2.647236969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.647236969\) |
| \(L(30)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + p^{29} T \) |
| good | 2 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 3 | \( 1 - 39759901988155 T + p^{58} T^{2} \) |
| 5 | \( 1 + 87370097058089297401 T + p^{58} T^{2} \) |
| 7 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 13 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 17 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 19 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 23 | \( 1 + \)\(35\!\cdots\!85\)\( T + p^{58} T^{2} \) |
| 29 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 31 | \( 1 - \)\(10\!\cdots\!83\)\( T + p^{58} T^{2} \) |
| 37 | \( 1 - \)\(32\!\cdots\!75\)\( T + p^{58} T^{2} \) |
| 41 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 43 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 47 | \( 1 + \)\(32\!\cdots\!50\)\( T + p^{58} T^{2} \) |
| 53 | \( 1 + \)\(17\!\cdots\!30\)\( T + p^{58} T^{2} \) |
| 59 | \( 1 - \)\(45\!\cdots\!47\)\( T + p^{58} T^{2} \) |
| 61 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 67 | \( 1 - \)\(17\!\cdots\!15\)\( T + p^{58} T^{2} \) |
| 71 | \( 1 - \)\(56\!\cdots\!87\)\( T + p^{58} T^{2} \) |
| 73 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 79 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 83 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 89 | \( 1 - \)\(57\!\cdots\!43\)\( T + p^{58} T^{2} \) |
| 97 | \( 1 - \)\(63\!\cdots\!55\)\( T + p^{58} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.99018664631107727929323098777, −9.776170659142957917379237009412, −8.245919826399157605532462364619, −7.69007099177359345747927927050, −6.38870565630682831569919212818, −5.27721283219955661152656437599, −3.72414193368419514718738848235, −2.76929112192737650972521602785, −2.05684774907154032102759919290, −0.60495985511287330863630376348,
0.60495985511287330863630376348, 2.05684774907154032102759919290, 2.76929112192737650972521602785, 3.72414193368419514718738848235, 5.27721283219955661152656437599, 6.38870565630682831569919212818, 7.69007099177359345747927927050, 8.245919826399157605532462364619, 9.776170659142957917379237009412, 10.99018664631107727929323098777
