| L(s) = 1 | + 3-s − 2.04·5-s + 1.81·7-s + 9-s − 1.62·11-s − 3.43·13-s − 2.04·15-s − 4.77·17-s + 5.04·19-s + 1.81·21-s + 3.71·23-s − 0.817·25-s + 27-s + 5.69·29-s + 4.23·31-s − 1.62·33-s − 3.70·35-s − 2.11·37-s − 3.43·39-s + 2.68·41-s + 12.6·43-s − 2.04·45-s − 8.44·47-s − 3.71·49-s − 4.77·51-s − 2.62·53-s + 3.31·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.914·5-s + 0.685·7-s + 0.333·9-s − 0.488·11-s − 0.952·13-s − 0.528·15-s − 1.15·17-s + 1.15·19-s + 0.395·21-s + 0.775·23-s − 0.163·25-s + 0.192·27-s + 1.05·29-s + 0.760·31-s − 0.282·33-s − 0.626·35-s − 0.347·37-s − 0.550·39-s + 0.419·41-s + 1.93·43-s − 0.304·45-s − 1.23·47-s − 0.530·49-s − 0.668·51-s − 0.360·53-s + 0.447·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.906089468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.906089468\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 2.04T + 5T^{2} \) |
| 7 | \( 1 - 1.81T + 7T^{2} \) |
| 11 | \( 1 + 1.62T + 11T^{2} \) |
| 13 | \( 1 + 3.43T + 13T^{2} \) |
| 17 | \( 1 + 4.77T + 17T^{2} \) |
| 19 | \( 1 - 5.04T + 19T^{2} \) |
| 23 | \( 1 - 3.71T + 23T^{2} \) |
| 29 | \( 1 - 5.69T + 29T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 + 2.11T + 37T^{2} \) |
| 41 | \( 1 - 2.68T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 + 8.44T + 47T^{2} \) |
| 53 | \( 1 + 2.62T + 53T^{2} \) |
| 59 | \( 1 - 0.400T + 59T^{2} \) |
| 61 | \( 1 - 0.131T + 61T^{2} \) |
| 67 | \( 1 - 6.63T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 2.93T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 5.43T + 89T^{2} \) |
| 97 | \( 1 - 9.62T + 97T^{2} \) |
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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
−8.152823392420379904527349093293, −7.915637805080853550742736142686, −7.18606494111998267865458431797, −6.45123910746357110811148459397, −5.09757676422802608213456293968, −4.75005924527812708853485540055, −3.84540394467493300297993478015, −2.91125211911210399897993211599, −2.15587107647667761396921754907, −0.76244571951829479981226224583,
0.76244571951829479981226224583, 2.15587107647667761396921754907, 2.91125211911210399897993211599, 3.84540394467493300297993478015, 4.75005924527812708853485540055, 5.09757676422802608213456293968, 6.45123910746357110811148459397, 7.18606494111998267865458431797, 7.915637805080853550742736142686, 8.152823392420379904527349093293
