L(s) = 1 | − 2.61·2-s − 3-s + 4.85·4-s + 0.236·5-s + 2.61·6-s − 4.23·7-s − 7.47·8-s + 9-s − 0.618·10-s − 5.23·11-s − 4.85·12-s − 3.23·13-s + 11.0·14-s − 0.236·15-s + 9.85·16-s + 2.47·17-s − 2.61·18-s − 1.76·19-s + 1.14·20-s + 4.23·21-s + 13.7·22-s + 3.23·23-s + 7.47·24-s − 4.94·25-s + 8.47·26-s − 27-s − 20.5·28-s + ⋯ |
L(s) = 1 | − 1.85·2-s − 0.577·3-s + 2.42·4-s + 0.105·5-s + 1.06·6-s − 1.60·7-s − 2.64·8-s + 0.333·9-s − 0.195·10-s − 1.57·11-s − 1.40·12-s − 0.897·13-s + 2.96·14-s − 0.0609·15-s + 2.46·16-s + 0.599·17-s − 0.617·18-s − 0.404·19-s + 0.256·20-s + 0.924·21-s + 2.92·22-s + 0.674·23-s + 1.52·24-s − 0.988·25-s + 1.66·26-s − 0.192·27-s − 3.88·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 5 | \( 1 - 0.236T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 + 6.70T + 41T^{2} \) |
| 43 | \( 1 - 3.70T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 + 1.23T + 73T^{2} \) |
| 79 | \( 1 + 0.472T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 9T + 97T^{2} \) |
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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
−13.11280807102653637204140655201, −12.20958483103405601939915029202, −10.86571364150307016472537442948, −10.03053886868186514601515193887, −9.446206329583624367338010042651, −7.908471012343790082513713795375, −6.96212014036167050528794276049, −5.76260028748807662172499793565, −2.71523764769336699533400004892, 0,
2.71523764769336699533400004892, 5.76260028748807662172499793565, 6.96212014036167050528794276049, 7.908471012343790082513713795375, 9.446206329583624367338010042651, 10.03053886868186514601515193887, 10.86571364150307016472537442948, 12.20958483103405601939915029202, 13.11280807102653637204140655201