| L(s) = 1 | − 1.06·2-s − 3-s − 0.857·4-s − 2.69·5-s + 1.06·6-s + 3.05·8-s + 9-s + 2.87·10-s + 11-s + 0.857·12-s − 3.28·13-s + 2.69·15-s − 1.54·16-s − 1.06·17-s − 1.06·18-s − 6.35·19-s + 2.30·20-s − 1.06·22-s − 1.54·23-s − 3.05·24-s + 2.24·25-s + 3.51·26-s − 27-s − 8.57·29-s − 2.87·30-s + 3.85·31-s − 4.45·32-s + ⋯ |
| L(s) = 1 | − 0.755·2-s − 0.577·3-s − 0.428·4-s − 1.20·5-s + 0.436·6-s + 1.07·8-s + 0.333·9-s + 0.909·10-s + 0.301·11-s + 0.247·12-s − 0.912·13-s + 0.695·15-s − 0.387·16-s − 0.259·17-s − 0.251·18-s − 1.45·19-s + 0.516·20-s − 0.227·22-s − 0.323·23-s − 0.623·24-s + 0.449·25-s + 0.689·26-s − 0.192·27-s − 1.59·29-s − 0.525·30-s + 0.692·31-s − 0.787·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2759168782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2759168782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.06T + 2T^{2} \) |
| 5 | \( 1 + 2.69T + 5T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 17 | \( 1 + 1.06T + 17T^{2} \) |
| 19 | \( 1 + 6.35T + 19T^{2} \) |
| 23 | \( 1 + 1.54T + 23T^{2} \) |
| 29 | \( 1 + 8.57T + 29T^{2} \) |
| 31 | \( 1 - 3.85T + 31T^{2} \) |
| 37 | \( 1 - 7.82T + 37T^{2} \) |
| 41 | \( 1 + 6.87T + 41T^{2} \) |
| 43 | \( 1 + 2.76T + 43T^{2} \) |
| 47 | \( 1 + 9.13T + 47T^{2} \) |
| 53 | \( 1 + 3.55T + 53T^{2} \) |
| 59 | \( 1 + 1.54T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 8.10T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 - 9.38T + 79T^{2} \) |
| 83 | \( 1 - 2.46T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−9.485395307890422454718744262486, −8.447889324365409297197058094786, −7.974544105540625381466893165590, −7.19133775329392629104493699047, −6.37100278588539083871462602442, −5.08536215187776815936593942046, −4.39150859602978975177093144309, −3.68101002497893153937652906654, −1.98140764730261505827659103463, −0.41382769098610629674117192113,
0.41382769098610629674117192113, 1.98140764730261505827659103463, 3.68101002497893153937652906654, 4.39150859602978975177093144309, 5.08536215187776815936593942046, 6.37100278588539083871462602442, 7.19133775329392629104493699047, 7.974544105540625381466893165590, 8.447889324365409297197058094786, 9.485395307890422454718744262486
