| L(s) = 1 | + 1.72e3·2-s − 5.90e4·3-s + 8.79e5·4-s − 1.01e8·6-s − 4.99e8·7-s − 2.10e9·8-s + 3.48e9·9-s − 1.14e11·11-s − 5.19e10·12-s − 1.56e11·13-s − 8.62e11·14-s − 5.46e12·16-s + 2.48e12·17-s + 6.01e12·18-s − 4.52e13·19-s + 2.95e13·21-s − 1.97e14·22-s − 6.05e12·23-s + 1.24e14·24-s − 2.69e14·26-s − 2.05e14·27-s − 4.39e14·28-s − 2.25e15·29-s + 2.57e15·31-s − 5.03e15·32-s + 6.75e15·33-s + 4.29e15·34-s + ⋯ |
| L(s) = 1 | + 1.19·2-s − 0.577·3-s + 0.419·4-s − 0.687·6-s − 0.669·7-s − 0.691·8-s + 0.333·9-s − 1.32·11-s − 0.242·12-s − 0.314·13-s − 0.797·14-s − 1.24·16-s + 0.299·17-s + 0.397·18-s − 1.69·19-s + 0.386·21-s − 1.58·22-s − 0.0304·23-s + 0.399·24-s − 0.374·26-s − 0.192·27-s − 0.280·28-s − 0.993·29-s + 0.564·31-s − 0.789·32-s + 0.767·33-s + 0.356·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.6685627947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6685627947\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.72e3T + 2.09e6T^{2} \) |
| 7 | \( 1 + 4.99e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.14e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 1.56e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 2.48e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 4.52e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 6.05e12T + 3.94e28T^{2} \) |
| 29 | \( 1 + 2.25e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 2.57e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 2.36e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 7.10e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.38e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 5.07e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.50e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 2.27e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 5.10e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.36e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 1.27e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 2.08e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 3.58e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.35e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 1.87e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 6.13e20T + 5.27e41T^{2} \) |
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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
−10.80225810850192281020412725977, −9.833639423183638194569553860111, −8.441827949293717229785530174485, −6.99422742547256855470357493893, −6.00151187028370473104679797716, −5.19711766772550606850431091634, −4.25236016141841075857633912976, −3.15776416608912715301114158597, −2.12911228690545830844956581896, −0.27028682365900038080695486010,
0.27028682365900038080695486010, 2.12911228690545830844956581896, 3.15776416608912715301114158597, 4.25236016141841075857633912976, 5.19711766772550606850431091634, 6.00151187028370473104679797716, 6.99422742547256855470357493893, 8.441827949293717229785530174485, 9.833639423183638194569553860111, 10.80225810850192281020412725977
