| L(s) = 1 | + 1.82·2-s + 2.31·3-s + 1.32·4-s + 4.21·6-s + 1.45·7-s − 1.23·8-s + 2.35·9-s − 3.89·11-s + 3.05·12-s + 3.05·13-s + 2.64·14-s − 4.89·16-s + 3.92·17-s + 4.29·18-s − 19-s + 3.35·21-s − 7.10·22-s − 5.37·23-s − 2.86·24-s + 5.57·26-s − 1.48·27-s + 1.91·28-s + 6·29-s − 8.43·31-s − 6.45·32-s − 9.01·33-s + 7.14·34-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 1.33·3-s + 0.660·4-s + 1.72·6-s + 0.548·7-s − 0.437·8-s + 0.785·9-s − 1.17·11-s + 0.883·12-s + 0.848·13-s + 0.706·14-s − 1.22·16-s + 0.951·17-s + 1.01·18-s − 0.229·19-s + 0.732·21-s − 1.51·22-s − 1.12·23-s − 0.584·24-s + 1.09·26-s − 0.286·27-s + 0.362·28-s + 1.11·29-s − 1.51·31-s − 1.14·32-s − 1.56·33-s + 1.22·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.696688056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.696688056\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.82T + 2T^{2} \) |
| 3 | \( 1 - 2.31T + 3T^{2} \) |
| 7 | \( 1 - 1.45T + 7T^{2} \) |
| 11 | \( 1 + 3.89T + 11T^{2} \) |
| 13 | \( 1 - 3.05T + 13T^{2} \) |
| 17 | \( 1 - 3.92T + 17T^{2} \) |
| 23 | \( 1 + 5.37T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8.43T + 31T^{2} \) |
| 37 | \( 1 + 5.95T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 1.45T + 43T^{2} \) |
| 47 | \( 1 - 4.90T + 47T^{2} \) |
| 53 | \( 1 + 4.23T + 53T^{2} \) |
| 59 | \( 1 - 3.35T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 9.84T + 67T^{2} \) |
| 71 | \( 1 - 8.64T + 71T^{2} \) |
| 73 | \( 1 + 2.43T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 3.05T + 97T^{2} \) |
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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
−11.15824548433602982690860490768, −10.15175496123620914402023416756, −9.014293372936536785430948776950, −8.244988940593869067741280650167, −7.50390831799230428673402923786, −6.04040376453593740695364213529, −5.16646578318485492417535539239, −4.00445050697294118143489854704, −3.20032577494930613732152515152, −2.15062333303143639152485539722,
2.15062333303143639152485539722, 3.20032577494930613732152515152, 4.00445050697294118143489854704, 5.16646578318485492417535539239, 6.04040376453593740695364213529, 7.50390831799230428673402923786, 8.244988940593869067741280650167, 9.014293372936536785430948776950, 10.15175496123620914402023416756, 11.15824548433602982690860490768
