| L(s) = 1 | − 2·2-s − 7·3-s − 4·4-s + 14·6-s + 24·8-s + 22·9-s − 5·11-s + 28·12-s + 14·13-s − 16·16-s + 21·17-s − 44·18-s + 49·19-s + 10·22-s + 159·23-s − 168·24-s − 28·26-s + 35·27-s + 58·29-s + 147·31-s − 160·32-s + 35·33-s − 42·34-s − 88·36-s − 219·37-s − 98·38-s − 98·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·3-s − 1/2·4-s + 0.952·6-s + 1.06·8-s + 0.814·9-s − 0.137·11-s + 0.673·12-s + 0.298·13-s − 1/4·16-s + 0.299·17-s − 0.576·18-s + 0.591·19-s + 0.0969·22-s + 1.44·23-s − 1.42·24-s − 0.211·26-s + 0.249·27-s + 0.371·29-s + 0.851·31-s − 0.883·32-s + 0.184·33-s − 0.211·34-s − 0.407·36-s − 0.973·37-s − 0.418·38-s − 0.402·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6415422623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6415422623\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p T + p^{3} T^{2} \) |
| 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 5 T + p^{3} T^{2} \) |
| 13 | \( 1 - 14 T + p^{3} T^{2} \) |
| 17 | \( 1 - 21 T + p^{3} T^{2} \) |
| 19 | \( 1 - 49 T + p^{3} T^{2} \) |
| 23 | \( 1 - 159 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 - 147 T + p^{3} T^{2} \) |
| 37 | \( 1 + 219 T + p^{3} T^{2} \) |
| 41 | \( 1 - 350 T + p^{3} T^{2} \) |
| 43 | \( 1 - 124 T + p^{3} T^{2} \) |
| 47 | \( 1 + 525 T + p^{3} T^{2} \) |
| 53 | \( 1 + 303 T + p^{3} T^{2} \) |
| 59 | \( 1 + 105 T + p^{3} T^{2} \) |
| 61 | \( 1 + 413 T + p^{3} T^{2} \) |
| 67 | \( 1 + 415 T + p^{3} T^{2} \) |
| 71 | \( 1 + 432 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1113 T + p^{3} T^{2} \) |
| 79 | \( 1 + 103 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1092 T + p^{3} T^{2} \) |
| 89 | \( 1 + 329 T + p^{3} T^{2} \) |
| 97 | \( 1 - 882 T + p^{3} T^{2} \) |
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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
−9.426670899096046995389081015706, −8.630674526408709795922972263891, −7.73489603456633435589177423529, −6.89129458613586184860720202367, −5.97203881446494432309442318180, −5.09573170225879288630687384414, −4.52405717775668519001363602699, −3.15269322759508638450764792069, −1.36053051834082953432577510564, −0.55710508201265729610557254226,
0.55710508201265729610557254226, 1.36053051834082953432577510564, 3.15269322759508638450764792069, 4.52405717775668519001363602699, 5.09573170225879288630687384414, 5.97203881446494432309442318180, 6.89129458613586184860720202367, 7.73489603456633435589177423529, 8.630674526408709795922972263891, 9.426670899096046995389081015706
