| L(s) = 1 | − 2.95e21·3-s + 1.23e27·4-s + 3.56e37·7-s + 8.72e42·9-s − 3.65e48·12-s − 2.64e50·13-s + 1.53e54·16-s + 4.22e57·19-s − 1.05e59·21-s + 8.07e62·25-s − 2.57e64·27-s + 4.41e64·28-s + 2.09e66·31-s + 1.08e70·36-s − 3.20e70·37-s + 7.82e71·39-s + 5.12e73·43-s − 4.52e75·48-s − 1.01e76·49-s − 3.27e77·52-s − 1.24e79·57-s − 3.80e80·61-s + 3.11e80·63-s + 1.89e81·64-s + 2.81e82·67-s − 1.35e84·73-s − 2.38e84·75-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 0.333·7-s + 9-s − 12-s − 1.97·13-s + 16-s + 1.20·19-s − 0.333·21-s + 25-s − 27-s + 0.333·28-s + 0.162·31-s + 36-s − 0.865·37-s + 1.97·39-s + 1.59·43-s − 48-s − 0.888·49-s − 1.97·52-s − 1.20·57-s − 1.73·61-s + 0.333·63-s + 64-s + 1.88·67-s − 1.91·73-s − 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(91-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+45) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{91}{2})\) |
\(\approx\) |
\(2.004753785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.004753785\) |
| \(L(46)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{45} T \) |
| good | 2 | \( ( 1 - p^{45} T )( 1 + p^{45} T ) \) |
| 5 | \( ( 1 - p^{45} T )( 1 + p^{45} T ) \) |
| 7 | \( 1 - \)\(35\!\cdots\!86\)\( T + p^{90} T^{2} \) |
| 11 | \( ( 1 - p^{45} T )( 1 + p^{45} T ) \) |
| 13 | \( 1 + \)\(26\!\cdots\!86\)\( T + p^{90} T^{2} \) |
| 17 | \( ( 1 - p^{45} T )( 1 + p^{45} T ) \) |
| 19 | \( 1 - \)\(42\!\cdots\!98\)\( T + p^{90} T^{2} \) |
| 23 | \( ( 1 - p^{45} T )( 1 + p^{45} T ) \) |
| 29 | \( ( 1 - p^{45} T )( 1 + p^{45} T ) \) |
| 31 | \( 1 - \)\(20\!\cdots\!02\)\( T + p^{90} T^{2} \) |
| 37 | \( 1 + \)\(32\!\cdots\!14\)\( T + p^{90} T^{2} \) |
| 41 | \( ( 1 - p^{45} T )( 1 + p^{45} T ) \) |
| 43 | \( 1 - \)\(51\!\cdots\!14\)\( T + p^{90} T^{2} \) |
| 47 | \( ( 1 - p^{45} T )( 1 + p^{45} T ) \) |
| 53 | \( ( 1 - p^{45} T )( 1 + p^{45} T ) \) |
| 59 | \( ( 1 - p^{45} T )( 1 + p^{45} T ) \) |
| 61 | \( 1 + \)\(38\!\cdots\!98\)\( T + p^{90} T^{2} \) |
| 67 | \( 1 - \)\(28\!\cdots\!86\)\( T + p^{90} T^{2} \) |
| 71 | \( ( 1 - p^{45} T )( 1 + p^{45} T ) \) |
| 73 | \( 1 + \)\(13\!\cdots\!86\)\( T + p^{90} T^{2} \) |
| 79 | \( 1 - \)\(30\!\cdots\!98\)\( T + p^{90} T^{2} \) |
| 83 | \( ( 1 - p^{45} T )( 1 + p^{45} T ) \) |
| 89 | \( ( 1 - p^{45} T )( 1 + p^{45} T ) \) |
| 97 | \( 1 - \)\(22\!\cdots\!86\)\( T + p^{90} T^{2} \) |
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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
−11.81586391540432032796525436169, −10.73906102792282579870227158578, −9.690450459147549654890122091265, −7.59692171932009035767348361693, −6.91418688375384195967778517359, −5.59374050573751075013662726193, −4.68986566045643498370839773172, −2.95187349861871910594524807273, −1.78223669187116952028897405459, −0.65872423527722920916889055159,
0.65872423527722920916889055159, 1.78223669187116952028897405459, 2.95187349861871910594524807273, 4.68986566045643498370839773172, 5.59374050573751075013662726193, 6.91418688375384195967778517359, 7.59692171932009035767348361693, 9.690450459147549654890122091265, 10.73906102792282579870227158578, 11.81586391540432032796525436169
