| L(s) = 1 | − 5.36e8·2-s + 5.72e13·3-s + 2.88e17·4-s − 3.07e22·6-s − 1.54e26·8-s − 1.42e27·9-s + 2.90e30·11-s + 1.65e31·12-s + 8.30e34·16-s + 9.55e35·17-s + 7.67e35·18-s − 1.58e37·19-s − 1.55e39·22-s − 8.86e39·24-s + 3.46e40·25-s − 3.51e41·27-s − 4.46e43·32-s + 1.66e44·33-s − 5.12e44·34-s − 4.11e44·36-s + 8.50e45·38-s + 1.17e47·41-s − 4.69e47·43-s + 8.36e47·44-s + 4.75e48·48-s + 1.03e49·49-s − 1.86e49·50-s + ⋯ |
| L(s) = 1 | − 2-s + 0.834·3-s + 4-s − 0.834·6-s − 8-s − 0.303·9-s + 1.82·11-s + 0.834·12-s + 16-s + 1.98·17-s + 0.303·18-s − 1.30·19-s − 1.82·22-s − 0.834·24-s + 25-s − 1.08·27-s − 32-s + 1.52·33-s − 1.98·34-s − 0.303·36-s + 1.30·38-s + 1.99·41-s − 1.99·43-s + 1.82·44-s + 0.834·48-s + 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(59-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+29) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{59}{2})\) |
\(\approx\) |
\(2.337986846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.337986846\) |
| \(L(30)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{29} T \) |
| good | 3 | \( 1 - 57281430144478 T + p^{58} T^{2} \) |
| 5 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 7 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 11 | \( 1 - \)\(29\!\cdots\!34\)\( T + p^{58} T^{2} \) |
| 13 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 17 | \( 1 - \)\(95\!\cdots\!82\)\( T + p^{58} T^{2} \) |
| 19 | \( 1 + \)\(15\!\cdots\!94\)\( T + p^{58} T^{2} \) |
| 23 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 29 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 31 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 37 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 41 | \( 1 - \)\(11\!\cdots\!34\)\( T + p^{58} T^{2} \) |
| 43 | \( 1 + \)\(46\!\cdots\!86\)\( T + p^{58} T^{2} \) |
| 47 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 53 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 59 | \( 1 - \)\(12\!\cdots\!78\)\( T + p^{58} T^{2} \) |
| 61 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 67 | \( 1 - \)\(17\!\cdots\!42\)\( T + p^{58} T^{2} \) |
| 71 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 73 | \( 1 + \)\(18\!\cdots\!02\)\( T + p^{58} T^{2} \) |
| 79 | \( ( 1 - p^{29} T )( 1 + p^{29} T ) \) |
| 83 | \( 1 + \)\(83\!\cdots\!22\)\( T + p^{58} T^{2} \) |
| 89 | \( 1 - \)\(25\!\cdots\!26\)\( T + p^{58} T^{2} \) |
| 97 | \( 1 + \)\(71\!\cdots\!34\)\( T + p^{58} T^{2} \) |
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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
−11.36427511966705549964364194011, −9.961755634853098925613625447751, −8.986763480118267898153900305213, −8.228961925062915691531175368537, −6.98708827960123739763752413280, −5.86455849734749675189359819885, −3.84541134475752884426038096525, −2.88215168247949750277422323166, −1.68302322424517124176436523655, −0.77144011132533528038947379081,
0.77144011132533528038947379081, 1.68302322424517124176436523655, 2.88215168247949750277422323166, 3.84541134475752884426038096525, 5.86455849734749675189359819885, 6.98708827960123739763752413280, 8.228961925062915691531175368537, 8.986763480118267898153900305213, 9.961755634853098925613625447751, 11.36427511966705549964364194011
