| L(s) = 1 | − 7·2-s − 38·3-s − 15·4-s + 266·6-s + 553·8-s + 715·9-s + 570·12-s + 1.08e3·13-s − 2.91e3·16-s − 5.00e3·18-s − 1.21e4·23-s − 2.10e4·24-s + 1.56e4·25-s − 7.57e3·26-s + 532·27-s + 3.07e4·29-s + 5.87e4·31-s − 1.50e4·32-s − 1.07e4·36-s − 4.11e4·39-s + 4.36e4·41-s + 8.51e4·46-s − 2.05e5·47-s + 1.10e5·48-s + 1.17e5·49-s − 1.09e5·50-s − 1.62e4·52-s + ⋯ |
| L(s) = 1 | − 7/8·2-s − 1.40·3-s − 0.234·4-s + 1.23·6-s + 1.08·8-s + 0.980·9-s + 0.329·12-s + 0.492·13-s − 0.710·16-s − 0.858·18-s − 23-s − 1.52·24-s + 25-s − 0.430·26-s + 0.0270·27-s + 1.26·29-s + 1.97·31-s − 0.458·32-s − 0.229·36-s − 0.693·39-s + 0.633·41-s + 7/8·46-s − 1.97·47-s + 1.00·48-s + 49-s − 7/8·50-s − 0.115·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4880303151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4880303151\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + p^{3} T \) |
| good | 2 | \( 1 + 7 T + p^{6} T^{2} \) |
| 3 | \( 1 + 38 T + p^{6} T^{2} \) |
| 5 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 7 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( 1 - 1082 T + p^{6} T^{2} \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( 1 - 30746 T + p^{6} T^{2} \) |
| 31 | \( 1 - 58754 T + p^{6} T^{2} \) |
| 37 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 41 | \( 1 - 43634 T + p^{6} T^{2} \) |
| 43 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 47 | \( 1 + 205342 T + p^{6} T^{2} \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( 1 + 253942 T + p^{6} T^{2} \) |
| 61 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 67 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 71 | \( 1 - 667154 T + p^{6} T^{2} \) |
| 73 | \( 1 - 725042 T + p^{6} T^{2} \) |
| 79 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
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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
−16.85679340562610487422197990914, −15.89194571733631421976646400420, −13.85616011152672178118138225480, −12.31744179707429698461728489589, −10.99805223113660726529032535557, −9.963420897852036337490473396685, −8.290094947978572616300124743303, −6.44575988406637144103056549585, −4.74761605237732684069464542721, −0.815114495854622862890383705685,
0.815114495854622862890383705685, 4.74761605237732684069464542721, 6.44575988406637144103056549585, 8.290094947978572616300124743303, 9.963420897852036337490473396685, 10.99805223113660726529032535557, 12.31744179707429698461728489589, 13.85616011152672178118138225480, 15.89194571733631421976646400420, 16.85679340562610487422197990914
