| L(s) = 1 | + 8.72e42·3-s + 1.53e54·4-s − 2.16e76·7-s + 7.61e85·9-s + 1.33e97·12-s + 3.41e100·13-s + 2.34e108·16-s − 6.59e114·19-s − 1.88e119·21-s + 6.52e125·25-s + 6.64e128·27-s − 3.31e130·28-s − 3.29e134·31-s + 1.16e140·36-s − 1.71e141·37-s + 2.97e143·39-s + 5.68e146·43-s + 2.04e151·48-s + 3.36e152·49-s + 5.23e154·52-s − 5.75e157·57-s + 4.88e160·61-s − 1.64e162·63-s + 3.59e162·64-s + 3.45e164·67-s + 8.40e167·73-s + 5.69e168·75-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s − 1.88·7-s + 9-s + 12-s + 1.89·13-s + 16-s − 0.538·19-s − 1.88·21-s + 25-s + 27-s − 1.88·28-s − 1.97·31-s + 36-s − 1.25·37-s + 1.89·39-s + 0.552·43-s + 48-s + 2.56·49-s + 1.89·52-s − 0.538·57-s + 1.02·61-s − 1.88·63-s + 64-s + 1.55·67-s + 1.68·73-s + 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(181-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+90) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{181}{2})\) |
\(\approx\) |
\(4.677240493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.677240493\) |
| \(L(91)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{90} T \) |
| good | 2 | \( ( 1 - p^{90} T )( 1 + p^{90} T ) \) |
| 5 | \( ( 1 - p^{90} T )( 1 + p^{90} T ) \) |
| 7 | \( 1 + \)\(21\!\cdots\!02\)\( T + p^{180} T^{2} \) |
| 11 | \( ( 1 - p^{90} T )( 1 + p^{90} T ) \) |
| 13 | \( 1 - \)\(34\!\cdots\!98\)\( T + p^{180} T^{2} \) |
| 17 | \( ( 1 - p^{90} T )( 1 + p^{90} T ) \) |
| 19 | \( 1 + \)\(65\!\cdots\!98\)\( T + p^{180} T^{2} \) |
| 23 | \( ( 1 - p^{90} T )( 1 + p^{90} T ) \) |
| 29 | \( ( 1 - p^{90} T )( 1 + p^{90} T ) \) |
| 31 | \( 1 + \)\(32\!\cdots\!98\)\( T + p^{180} T^{2} \) |
| 37 | \( 1 + \)\(17\!\cdots\!02\)\( T + p^{180} T^{2} \) |
| 41 | \( ( 1 - p^{90} T )( 1 + p^{90} T ) \) |
| 43 | \( 1 - \)\(56\!\cdots\!98\)\( T + p^{180} T^{2} \) |
| 47 | \( ( 1 - p^{90} T )( 1 + p^{90} T ) \) |
| 53 | \( ( 1 - p^{90} T )( 1 + p^{90} T ) \) |
| 59 | \( ( 1 - p^{90} T )( 1 + p^{90} T ) \) |
| 61 | \( 1 - \)\(48\!\cdots\!02\)\( T + p^{180} T^{2} \) |
| 67 | \( 1 - \)\(34\!\cdots\!98\)\( T + p^{180} T^{2} \) |
| 71 | \( ( 1 - p^{90} T )( 1 + p^{90} T ) \) |
| 73 | \( 1 - \)\(84\!\cdots\!98\)\( T + p^{180} T^{2} \) |
| 79 | \( 1 + \)\(12\!\cdots\!98\)\( T + p^{180} T^{2} \) |
| 83 | \( ( 1 - p^{90} T )( 1 + p^{90} T ) \) |
| 89 | \( ( 1 - p^{90} T )( 1 + p^{90} T ) \) |
| 97 | \( 1 + \)\(77\!\cdots\!02\)\( T + p^{180} T^{2} \) |
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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
−9.195272872273674093436945339487, −8.438547411347403973196079109797, −7.09883763991717723726350936734, −6.60112089843965673280633490637, −5.69718521184109916098270840560, −3.74642284165599933860880545539, −3.47684504387764523253108666637, −2.58041838955465472896306332927, −1.65561282044800022609231947957, −0.68147329140862907038493063619,
0.68147329140862907038493063619, 1.65561282044800022609231947957, 2.58041838955465472896306332927, 3.47684504387764523253108666637, 3.74642284165599933860880545539, 5.69718521184109916098270840560, 6.60112089843965673280633490637, 7.09883763991717723726350936734, 8.438547411347403973196079109797, 9.195272872273674093436945339487
