| L(s) = 1 | + 0.902·2-s − 3-s − 1.18·4-s − 5-s − 0.902·6-s − 0.451·7-s − 2.87·8-s + 9-s − 0.902·10-s + 5.51·11-s + 1.18·12-s + 5.84·13-s − 0.407·14-s + 15-s − 0.225·16-s + 5.83·17-s + 0.902·18-s + 0.208·19-s + 1.18·20-s + 0.451·21-s + 4.97·22-s + 2.87·24-s + 25-s + 5.27·26-s − 27-s + 0.534·28-s + 8.20·29-s + ⋯ |
| L(s) = 1 | + 0.638·2-s − 0.577·3-s − 0.592·4-s − 0.447·5-s − 0.368·6-s − 0.170·7-s − 1.01·8-s + 0.333·9-s − 0.285·10-s + 1.66·11-s + 0.342·12-s + 1.62·13-s − 0.108·14-s + 0.258·15-s − 0.0562·16-s + 1.41·17-s + 0.212·18-s + 0.0477·19-s + 0.265·20-s + 0.0984·21-s + 1.06·22-s + 0.586·24-s + 0.200·25-s + 1.03·26-s − 0.192·27-s + 0.101·28-s + 1.52·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.163242867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.163242867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.902T + 2T^{2} \) |
| 7 | \( 1 + 0.451T + 7T^{2} \) |
| 11 | \( 1 - 5.51T + 11T^{2} \) |
| 13 | \( 1 - 5.84T + 13T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 19 | \( 1 - 0.208T + 19T^{2} \) |
| 29 | \( 1 - 8.20T + 29T^{2} \) |
| 31 | \( 1 + 1.38T + 31T^{2} \) |
| 37 | \( 1 - 2.33T + 37T^{2} \) |
| 41 | \( 1 + 4.76T + 41T^{2} \) |
| 43 | \( 1 + 3.59T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 9.73T + 53T^{2} \) |
| 59 | \( 1 - 5.24T + 59T^{2} \) |
| 61 | \( 1 + 3.07T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 7.39T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 1.08T + 79T^{2} \) |
| 83 | \( 1 - 7.73T + 83T^{2} \) |
| 89 | \( 1 + 0.255T + 89T^{2} \) |
| 97 | \( 1 - 1.97T + 97T^{2} \) |
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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
−7.936194051988483713937715026025, −6.80068862434926826320678894405, −6.34018370268785576920095841643, −5.80004997480174915432313881822, −4.95415042182166671378208290829, −4.29164279465962171919634425819, −3.53559326286228602591578178077, −3.27421540568356158744008443261, −1.45843256787291422377819940524, −0.77615209877290124926074890456,
0.77615209877290124926074890456, 1.45843256787291422377819940524, 3.27421540568356158744008443261, 3.53559326286228602591578178077, 4.29164279465962171919634425819, 4.95415042182166671378208290829, 5.80004997480174915432313881822, 6.34018370268785576920095841643, 6.80068862434926826320678894405, 7.936194051988483713937715026025
