L(s) = 1 | − 1.61·2-s − 3-s + 0.618·4-s + 1.61·6-s − 1.61·7-s + 2.23·8-s + 9-s + 4.23·11-s − 0.618·12-s − 1.76·13-s + 2.61·14-s − 4.85·16-s − 4.38·17-s − 1.61·18-s − 5·19-s + 1.61·21-s − 6.85·22-s + 5.47·23-s − 2.23·24-s + 2.85·26-s − 27-s − 1.00·28-s − 6.70·29-s − 8.85·31-s + 3.38·32-s − 4.23·33-s + 7.09·34-s + ⋯ |
L(s) = 1 | − 1.14·2-s − 0.577·3-s + 0.309·4-s + 0.660·6-s − 0.611·7-s + 0.790·8-s + 0.333·9-s + 1.27·11-s − 0.178·12-s − 0.489·13-s + 0.699·14-s − 1.21·16-s − 1.06·17-s − 0.381·18-s − 1.14·19-s + 0.353·21-s − 1.46·22-s + 1.14·23-s − 0.456·24-s + 0.559·26-s − 0.192·27-s − 0.188·28-s − 1.24·29-s − 1.59·31-s + 0.597·32-s − 0.737·33-s + 1.21·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 + 4.38T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 + 8.85T + 31T^{2} \) |
| 37 | \( 1 - 4.23T + 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 7.09T + 53T^{2} \) |
| 59 | \( 1 - 6.38T + 59T^{2} \) |
| 61 | \( 1 + 4.38T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 1.85T + 73T^{2} \) |
| 79 | \( 1 - 1.70T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 3.94T + 89T^{2} \) |
| 97 | \( 1 + 18.8T + 97T^{2} \) |
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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
−10.92961887981939164886187471218, −9.748583248005561838571615477786, −9.254079003524244644984733378415, −8.366016686210491811455238587665, −7.02051547758287872334750586570, −6.54597364204454654091182398997, −5.00366836025177682806116353243, −3.84376720518480584167072850542, −1.78705715347479934852776939693, 0,
1.78705715347479934852776939693, 3.84376720518480584167072850542, 5.00366836025177682806116353243, 6.54597364204454654091182398997, 7.02051547758287872334750586570, 8.366016686210491811455238587665, 9.254079003524244644984733378415, 9.748583248005561838571615477786, 10.92961887981939164886187471218