| L(s) = 1 | + 77.0·2-s − 2.22e3·3-s − 2.24e3·4-s − 4.22e4·5-s − 1.71e5·6-s − 3.55e5·7-s − 8.04e5·8-s + 3.34e6·9-s − 3.25e6·10-s − 4.93e6·11-s + 4.99e6·12-s − 1.79e7·13-s − 2.74e7·14-s + 9.38e7·15-s − 4.36e7·16-s + 1.12e8·17-s + 2.57e8·18-s − 1.54e8·19-s + 9.50e7·20-s + 7.89e8·21-s − 3.80e8·22-s + 1.48e8·23-s + 1.78e9·24-s + 5.64e8·25-s − 1.38e9·26-s − 3.87e9·27-s + 7.99e8·28-s + ⋯ |
| L(s) = 1 | + 0.851·2-s − 1.75·3-s − 0.274·4-s − 1.20·5-s − 1.49·6-s − 1.14·7-s − 1.08·8-s + 2.09·9-s − 1.03·10-s − 0.840·11-s + 0.482·12-s − 1.03·13-s − 0.972·14-s + 2.12·15-s − 0.650·16-s + 1.13·17-s + 1.78·18-s − 0.751·19-s + 0.331·20-s + 2.00·21-s − 0.715·22-s + 0.208·23-s + 1.90·24-s + 0.462·25-s − 0.878·26-s − 1.92·27-s + 0.313·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.002492338825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002492338825\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 1.48e8T \) |
| good | 2 | \( 1 - 77.0T + 8.19e3T^{2} \) |
| 3 | \( 1 + 2.22e3T + 1.59e6T^{2} \) |
| 5 | \( 1 + 4.22e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 3.55e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 4.93e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 1.79e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.12e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.54e8T + 4.20e16T^{2} \) |
| 29 | \( 1 + 6.25e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 3.75e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 1.27e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 4.64e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 2.40e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 1.14e11T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.00e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 3.80e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 1.53e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 6.35e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.63e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.26e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 3.10e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.16e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 7.35e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.94e12T + 6.73e25T^{2} \) |
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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
−14.97489162942097104523043589330, −12.81661010778623843110634336605, −12.45424423857137739357791521939, −11.25454232547131576507283366137, −9.808816091978303217483152491470, −7.38716213385635637711477456969, −5.91400911727640189387479376956, −4.84711226255244368275714612126, −3.53566010939778389115751710871, −0.02969013873710292711104325466,
0.02969013873710292711104325466, 3.53566010939778389115751710871, 4.84711226255244368275714612126, 5.91400911727640189387479376956, 7.38716213385635637711477456969, 9.808816091978303217483152491470, 11.25454232547131576507283366137, 12.45424423857137739357791521939, 12.81661010778623843110634336605, 14.97489162942097104523043589330
