L(s) = 1 | + 1.55·2-s − 3-s + 0.409·4-s − 1.55·6-s − 2.46·8-s + 9-s + 4.43·11-s − 0.409·12-s − 1.73·13-s − 4.65·16-s + 2.73·17-s + 1.55·18-s − 0.305·19-s + 6.87·22-s + 7.02·23-s + 2.46·24-s − 2.68·26-s − 27-s − 7.79·29-s − 5.28·31-s − 2.28·32-s − 4.43·33-s + 4.24·34-s + 0.409·36-s + 3.67·37-s − 0.473·38-s + 1.73·39-s + ⋯ |
L(s) = 1 | + 1.09·2-s − 0.577·3-s + 0.204·4-s − 0.633·6-s − 0.872·8-s + 0.333·9-s + 1.33·11-s − 0.118·12-s − 0.480·13-s − 1.16·16-s + 0.662·17-s + 0.365·18-s − 0.0700·19-s + 1.46·22-s + 1.46·23-s + 0.503·24-s − 0.527·26-s − 0.192·27-s − 1.44·29-s − 0.949·31-s − 0.403·32-s − 0.771·33-s + 0.727·34-s + 0.0683·36-s + 0.604·37-s − 0.0768·38-s + 0.277·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.470453658\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.470453658\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.55T + 2T^{2} \) |
| 11 | \( 1 - 4.43T + 11T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 - 2.73T + 17T^{2} \) |
| 19 | \( 1 + 0.305T + 19T^{2} \) |
| 23 | \( 1 - 7.02T + 23T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 37 | \( 1 - 3.67T + 37T^{2} \) |
| 41 | \( 1 + 6.71T + 41T^{2} \) |
| 43 | \( 1 - 9.71T + 43T^{2} \) |
| 47 | \( 1 + 1.81T + 47T^{2} \) |
| 53 | \( 1 - 1.71T + 53T^{2} \) |
| 59 | \( 1 + 1.14T + 59T^{2} \) |
| 61 | \( 1 - 9.55T + 61T^{2} \) |
| 67 | \( 1 - 8.38T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 6.71T + 79T^{2} \) |
| 83 | \( 1 + 5.09T + 83T^{2} \) |
| 89 | \( 1 - 4.07T + 89T^{2} \) |
| 97 | \( 1 - 2.87T + 97T^{2} \) |
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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
−8.658528600001377641675778585066, −7.47690086429382286639423374436, −6.83872161429694912600176428899, −6.12094499073450481535944342163, −5.38212628718924492370288654333, −4.85080007554315219258554098273, −3.88695912529639944518670751471, −3.43065365460216423628195190383, −2.15134617076524180525029447010, −0.813113504252578828712040397866,
0.813113504252578828712040397866, 2.15134617076524180525029447010, 3.43065365460216423628195190383, 3.88695912529639944518670751471, 4.85080007554315219258554098273, 5.38212628718924492370288654333, 6.12094499073450481535944342163, 6.83872161429694912600176428899, 7.47690086429382286639423374436, 8.658528600001377641675778585066