| L(s) = 1 | − 1.57e27·3-s + 2.07e34·4-s + 2.81e48·7-s + 2.46e54·9-s − 3.26e61·12-s − 5.07e63·13-s + 4.31e68·16-s − 1.32e73·19-s − 4.41e75·21-s + 4.81e79·25-s − 3.87e81·27-s + 5.84e82·28-s + 1.02e85·31-s + 5.11e88·36-s + 4.87e89·37-s + 7.97e90·39-s − 2.10e93·43-s − 6.77e95·48-s + 5.72e96·49-s − 1.05e98·52-s + 2.08e100·57-s − 6.07e101·61-s + 6.93e102·63-s + 8.95e102·64-s + 1.65e104·67-s + 1.77e106·73-s − 7.55e106·75-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 1.90·7-s + 9-s − 12-s − 1.62·13-s + 16-s − 1.71·19-s − 1.90·21-s + 25-s − 27-s + 1.90·28-s + 1.00·31-s + 36-s + 1.99·37-s + 1.62·39-s − 1.64·43-s − 48-s + 2.61·49-s − 1.62·52-s + 1.71·57-s − 1.04·61-s + 1.90·63-s + 64-s + 1.35·67-s + 1.09·73-s − 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(115-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+57) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{115}{2})\) |
\(\approx\) |
\(2.715992687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.715992687\) |
| \(L(58)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{57} T \) |
| good | 2 | \( ( 1 - p^{57} T )( 1 + p^{57} T ) \) |
| 5 | \( ( 1 - p^{57} T )( 1 + p^{57} T ) \) |
| 7 | \( 1 - \)\(28\!\cdots\!86\)\( T + p^{114} T^{2} \) |
| 11 | \( ( 1 - p^{57} T )( 1 + p^{57} T ) \) |
| 13 | \( 1 + \)\(50\!\cdots\!66\)\( T + p^{114} T^{2} \) |
| 17 | \( ( 1 - p^{57} T )( 1 + p^{57} T ) \) |
| 19 | \( 1 + \)\(13\!\cdots\!22\)\( T + p^{114} T^{2} \) |
| 23 | \( ( 1 - p^{57} T )( 1 + p^{57} T ) \) |
| 29 | \( ( 1 - p^{57} T )( 1 + p^{57} T ) \) |
| 31 | \( 1 - \)\(10\!\cdots\!22\)\( T + p^{114} T^{2} \) |
| 37 | \( 1 - \)\(48\!\cdots\!66\)\( T + p^{114} T^{2} \) |
| 41 | \( ( 1 - p^{57} T )( 1 + p^{57} T ) \) |
| 43 | \( 1 + \)\(21\!\cdots\!86\)\( T + p^{114} T^{2} \) |
| 47 | \( ( 1 - p^{57} T )( 1 + p^{57} T ) \) |
| 53 | \( ( 1 - p^{57} T )( 1 + p^{57} T ) \) |
| 59 | \( ( 1 - p^{57} T )( 1 + p^{57} T ) \) |
| 61 | \( 1 + \)\(60\!\cdots\!58\)\( T + p^{114} T^{2} \) |
| 67 | \( 1 - \)\(16\!\cdots\!46\)\( T + p^{114} T^{2} \) |
| 71 | \( ( 1 - p^{57} T )( 1 + p^{57} T ) \) |
| 73 | \( 1 - \)\(17\!\cdots\!94\)\( T + p^{114} T^{2} \) |
| 79 | \( 1 + \)\(21\!\cdots\!82\)\( T + p^{114} T^{2} \) |
| 83 | \( ( 1 - p^{57} T )( 1 + p^{57} T ) \) |
| 89 | \( ( 1 - p^{57} T )( 1 + p^{57} T ) \) |
| 97 | \( 1 - \)\(19\!\cdots\!26\)\( T + p^{114} T^{2} \) |
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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
−11.02772004511313614440869318837, −10.13206768094205047998090842582, −8.209424440313318597641618700441, −7.30210576666683055079721574535, −6.29334407445452801431444031452, −5.03995981743108066362276624641, −4.45209379762026638586827023204, −2.46200825618183443176703362698, −1.72844334371120606133474026114, −0.70547117562071532015905857353,
0.70547117562071532015905857353, 1.72844334371120606133474026114, 2.46200825618183443176703362698, 4.45209379762026638586827023204, 5.03995981743108066362276624641, 6.29334407445452801431444031452, 7.30210576666683055079721574535, 8.209424440313318597641618700441, 10.13206768094205047998090842582, 11.02772004511313614440869318837
