L(s) = 1 | + 2-s + 3-s + 4-s + 2.47·5-s + 6-s + 8-s + 9-s + 2.47·10-s + 11-s + 12-s − 3.03·13-s + 2.47·15-s + 16-s + 6.64·17-s + 18-s − 0.557·19-s + 2.47·20-s + 22-s − 2.66·23-s + 24-s + 1.11·25-s − 3.03·26-s + 27-s + 3.77·29-s + 2.47·30-s − 2.00·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.10·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.782·10-s + 0.301·11-s + 0.288·12-s − 0.840·13-s + 0.638·15-s + 0.250·16-s + 1.61·17-s + 0.235·18-s − 0.127·19-s + 0.553·20-s + 0.213·22-s − 0.556·23-s + 0.204·24-s + 0.223·25-s − 0.594·26-s + 0.192·27-s + 0.701·29-s + 0.451·30-s − 0.360·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.734159470\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.734159470\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 2.47T + 5T^{2} \) |
| 13 | \( 1 + 3.03T + 13T^{2} \) |
| 17 | \( 1 - 6.64T + 17T^{2} \) |
| 19 | \( 1 + 0.557T + 19T^{2} \) |
| 23 | \( 1 + 2.66T + 23T^{2} \) |
| 29 | \( 1 - 3.77T + 29T^{2} \) |
| 31 | \( 1 + 2.00T + 31T^{2} \) |
| 37 | \( 1 + 0.158T + 37T^{2} \) |
| 41 | \( 1 - 5.93T + 41T^{2} \) |
| 43 | \( 1 - 4.32T + 43T^{2} \) |
| 47 | \( 1 - 7.38T + 47T^{2} \) |
| 53 | \( 1 - 2.83T + 53T^{2} \) |
| 59 | \( 1 - 1.48T + 59T^{2} \) |
| 61 | \( 1 + 3.19T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 3.18T + 73T^{2} \) |
| 79 | \( 1 + 6.95T + 79T^{2} \) |
| 83 | \( 1 + 4.59T + 83T^{2} \) |
| 89 | \( 1 - 6.62T + 89T^{2} \) |
| 97 | \( 1 - 1.17T + 97T^{2} \) |
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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
−8.736835073337423715774084749825, −7.67346575048908598616393603458, −7.25824832336489705227293654810, −6.10868420100799061272829347893, −5.73416938212103050798985348375, −4.81108351675495564884440893313, −3.94997493676207939904723072580, −2.96989221621969619232300824323, −2.26410950740894070673604638126, −1.28143006507978761139222444456,
1.28143006507978761139222444456, 2.26410950740894070673604638126, 2.96989221621969619232300824323, 3.94997493676207939904723072580, 4.81108351675495564884440893313, 5.73416938212103050798985348375, 6.10868420100799061272829347893, 7.25824832336489705227293654810, 7.67346575048908598616393603458, 8.736835073337423715774084749825