L(s) = 1 | + 1.49·2-s − 3-s + 0.245·4-s − 1.49·6-s + 3.73·7-s − 2.62·8-s + 9-s + 4.52·11-s − 0.245·12-s + 6.66·13-s + 5.59·14-s − 4.43·16-s + 1.58·17-s + 1.49·18-s − 0.710·19-s − 3.73·21-s + 6.78·22-s − 8.96·23-s + 2.62·24-s + 9.98·26-s − 27-s + 0.917·28-s + 1.28·29-s + 6.01·31-s − 1.38·32-s − 4.52·33-s + 2.38·34-s + ⋯ |
L(s) = 1 | + 1.05·2-s − 0.577·3-s + 0.122·4-s − 0.611·6-s + 1.41·7-s − 0.929·8-s + 0.333·9-s + 1.36·11-s − 0.0709·12-s + 1.84·13-s + 1.49·14-s − 1.10·16-s + 0.385·17-s + 0.353·18-s − 0.162·19-s − 0.814·21-s + 1.44·22-s − 1.87·23-s + 0.536·24-s + 1.95·26-s − 0.192·27-s + 0.173·28-s + 0.238·29-s + 1.08·31-s − 0.244·32-s − 0.787·33-s + 0.408·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.301625581\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.301625581\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 - 1.49T + 2T^{2} \) |
| 7 | \( 1 - 3.73T + 7T^{2} \) |
| 11 | \( 1 - 4.52T + 11T^{2} \) |
| 13 | \( 1 - 6.66T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 + 0.710T + 19T^{2} \) |
| 23 | \( 1 + 8.96T + 23T^{2} \) |
| 29 | \( 1 - 1.28T + 29T^{2} \) |
| 31 | \( 1 - 6.01T + 31T^{2} \) |
| 37 | \( 1 - 4.78T + 37T^{2} \) |
| 41 | \( 1 + 7.22T + 41T^{2} \) |
| 43 | \( 1 + 4.67T + 43T^{2} \) |
| 53 | \( 1 + 2.17T + 53T^{2} \) |
| 59 | \( 1 - 4.30T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 9.91T + 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 - 7.73T + 83T^{2} \) |
| 89 | \( 1 + 9.77T + 89T^{2} \) |
| 97 | \( 1 - 7.66T + 97T^{2} \) |
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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
−8.346655088601347850245177372549, −8.047137663812662020113893479950, −6.53431635632346573353272160861, −6.29212277537480826346135515840, −5.47976692920959189831846319672, −4.69212731336331279121156011752, −4.03950625892774012982406951164, −3.51058559966936228698681372082, −1.93724195079232779647912257076, −1.03500544979135913362926746855,
1.03500544979135913362926746855, 1.93724195079232779647912257076, 3.51058559966936228698681372082, 4.03950625892774012982406951164, 4.69212731336331279121156011752, 5.47976692920959189831846319672, 6.29212277537480826346135515840, 6.53431635632346573353272160861, 8.047137663812662020113893479950, 8.346655088601347850245177372549