| L(s) = 1 | + 1.23·3-s − 4.08·5-s − 5.04·7-s − 1.47·9-s − 1.54·11-s − 1.19·13-s − 5.04·15-s − 5.18·17-s − 0.101·19-s − 6.23·21-s + 11.7·25-s − 5.52·27-s − 5.58·29-s − 7.41·31-s − 1.91·33-s + 20.6·35-s + 2.86·37-s − 1.47·39-s − 7.75·41-s − 4.13·43-s + 6.03·45-s + 2.43·47-s + 18.5·49-s − 6.39·51-s − 8.63·53-s + 6.33·55-s − 0.125·57-s + ⋯ |
| L(s) = 1 | + 0.712·3-s − 1.82·5-s − 1.90·7-s − 0.491·9-s − 0.467·11-s − 0.331·13-s − 1.30·15-s − 1.25·17-s − 0.0232·19-s − 1.36·21-s + 2.34·25-s − 1.06·27-s − 1.03·29-s − 1.33·31-s − 0.333·33-s + 3.49·35-s + 0.471·37-s − 0.236·39-s − 1.21·41-s − 0.631·43-s + 0.899·45-s + 0.354·47-s + 2.64·49-s − 0.896·51-s − 1.18·53-s + 0.854·55-s − 0.0165·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08963114141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08963114141\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 1.23T + 3T^{2} \) |
| 5 | \( 1 + 4.08T + 5T^{2} \) |
| 7 | \( 1 + 5.04T + 7T^{2} \) |
| 11 | \( 1 + 1.54T + 11T^{2} \) |
| 13 | \( 1 + 1.19T + 13T^{2} \) |
| 17 | \( 1 + 5.18T + 17T^{2} \) |
| 19 | \( 1 + 0.101T + 19T^{2} \) |
| 29 | \( 1 + 5.58T + 29T^{2} \) |
| 31 | \( 1 + 7.41T + 31T^{2} \) |
| 37 | \( 1 - 2.86T + 37T^{2} \) |
| 41 | \( 1 + 7.75T + 41T^{2} \) |
| 43 | \( 1 + 4.13T + 43T^{2} \) |
| 47 | \( 1 - 2.43T + 47T^{2} \) |
| 53 | \( 1 + 8.63T + 53T^{2} \) |
| 59 | \( 1 + 2.32T + 59T^{2} \) |
| 61 | \( 1 + 2.43T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 6.33T + 71T^{2} \) |
| 73 | \( 1 + 2.31T + 73T^{2} \) |
| 79 | \( 1 + 1.53T + 79T^{2} \) |
| 83 | \( 1 - 0.719T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 - 9.90T + 97T^{2} \) |
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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
−8.379978009868573203540753174600, −7.68937937188882763523400301426, −7.06575842302975495243223409701, −6.45323528418005380427820009667, −5.40822950666402157714288692941, −4.33536663051269822973884019540, −3.54550871115980426648097543110, −3.22251653360991735055330594901, −2.31639969047863981199789219222, −0.14859341265290506009171798570,
0.14859341265290506009171798570, 2.31639969047863981199789219222, 3.22251653360991735055330594901, 3.54550871115980426648097543110, 4.33536663051269822973884019540, 5.40822950666402157714288692941, 6.45323528418005380427820009667, 7.06575842302975495243223409701, 7.68937937188882763523400301426, 8.379978009868573203540753174600
