| L(s) = 1 | − 7.63·2-s − 15.6·3-s + 42.3·4-s + 119.·6-s − 201.·8-s + 162.·9-s − 661.·12-s + 215.·13-s + 859.·16-s − 1.24e3·18-s + 529·23-s + 3.14e3·24-s + 625·25-s − 1.64e3·26-s − 1.27e3·27-s + 115.·29-s − 1.13e3·31-s − 3.34e3·32-s + 6.89e3·36-s − 3.35e3·39-s + 2.26e3·41-s − 4.04e3·46-s + 4.39e3·47-s − 1.34e4·48-s + 2.40e3·49-s − 4.77e3·50-s + 9.10e3·52-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 1.73·3-s + 2.64·4-s + 3.31·6-s − 3.14·8-s + 2.01·9-s − 4.59·12-s + 1.27·13-s + 3.35·16-s − 3.84·18-s + 23-s + 5.45·24-s + 25-s − 2.42·26-s − 1.75·27-s + 0.137·29-s − 1.18·31-s − 3.26·32-s + 5.32·36-s − 2.20·39-s + 1.34·41-s − 1.90·46-s + 1.99·47-s − 5.82·48-s + 49-s − 1.90·50-s + 3.36·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3272874983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3272874983\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 529T \) |
| good | 2 | \( 1 + 7.63T + 16T^{2} \) |
| 3 | \( 1 + 15.6T + 81T^{2} \) |
| 5 | \( 1 - 625T^{2} \) |
| 7 | \( 1 - 2.40e3T^{2} \) |
| 11 | \( 1 - 1.46e4T^{2} \) |
| 13 | \( 1 - 215.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 8.35e4T^{2} \) |
| 19 | \( 1 - 1.30e5T^{2} \) |
| 29 | \( 1 - 115.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.13e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.26e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 3.41e6T^{2} \) |
| 47 | \( 1 - 4.39e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 7.89e6T^{2} \) |
| 59 | \( 1 + 6.28e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 1.38e7T^{2} \) |
| 67 | \( 1 - 2.01e7T^{2} \) |
| 71 | \( 1 - 9.77e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 7.42e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.89e7T^{2} \) |
| 83 | \( 1 - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.27e7T^{2} \) |
| 97 | \( 1 - 8.85e7T^{2} \) |
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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
−17.18508894776889709528440640136, −16.45661102891033285661071658872, −15.53422510220330115346818693819, −12.46128035379592200358755504161, −11.12683028791975103165187350270, −10.65164785317094487885278417037, −9.037673326943458348505686004887, −7.19055855135487678825848289663, −5.95903244809796321649116099652, −0.945467964969351434388544064891,
0.945467964969351434388544064891, 5.95903244809796321649116099652, 7.19055855135487678825848289663, 9.037673326943458348505686004887, 10.65164785317094487885278417037, 11.12683028791975103165187350270, 12.46128035379592200358755504161, 15.53422510220330115346818693819, 16.45661102891033285661071658872, 17.18508894776889709528440640136
