L(s) = 1 | − 2-s − 3-s + 4-s − 3.46·5-s + 6-s − 8-s + 9-s + 3.46·10-s + 0.278·11-s − 12-s + 6.92·13-s + 3.46·15-s + 16-s + 5.79·17-s − 18-s + 19-s − 3.46·20-s − 0.278·22-s + 3.09·23-s + 24-s + 6.98·25-s − 6.92·26-s − 27-s − 8.55·29-s − 3.46·30-s + 7.34·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.54·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.09·10-s + 0.0838·11-s − 0.288·12-s + 1.91·13-s + 0.893·15-s + 0.250·16-s + 1.40·17-s − 0.235·18-s + 0.229·19-s − 0.773·20-s − 0.0593·22-s + 0.644·23-s + 0.204·24-s + 1.39·25-s − 1.35·26-s − 0.192·27-s − 1.58·29-s − 0.631·30-s + 1.32·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9575941405\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9575941405\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 - 0.278T + 11T^{2} \) |
| 13 | \( 1 - 6.92T + 13T^{2} \) |
| 17 | \( 1 - 5.79T + 17T^{2} \) |
| 23 | \( 1 - 3.09T + 23T^{2} \) |
| 29 | \( 1 + 8.55T + 29T^{2} \) |
| 31 | \( 1 - 7.34T + 31T^{2} \) |
| 37 | \( 1 + 5.24T + 37T^{2} \) |
| 41 | \( 1 - 8.88T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 1.79T + 47T^{2} \) |
| 53 | \( 1 - 8.38T + 53T^{2} \) |
| 59 | \( 1 + 7.79T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 0.684T + 71T^{2} \) |
| 73 | \( 1 + 1.92T + 73T^{2} \) |
| 79 | \( 1 - 1.40T + 79T^{2} \) |
| 83 | \( 1 + 0.146T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 - 4.09T + 97T^{2} \) |
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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
−8.027154816023088368330648636145, −7.58305996509890733349703829200, −6.93108621189023595965324681089, −6.02018221672547255851959453500, −5.47462493426065146735221176690, −4.25828589944899078912211299922, −3.72127727765906776934097025754, −2.97045715824663710751863616726, −1.35122825452433132514110915833, −0.68652414094213973213125007465,
0.68652414094213973213125007465, 1.35122825452433132514110915833, 2.97045715824663710751863616726, 3.72127727765906776934097025754, 4.25828589944899078912211299922, 5.47462493426065146735221176690, 6.02018221672547255851959453500, 6.93108621189023595965324681089, 7.58305996509890733349703829200, 8.027154816023088368330648636145