| L(s) = 1 | + 3-s + 3.03·5-s + 9-s − 11-s − 6.20·13-s + 3.03·15-s − 3.89·17-s − 4.76·19-s − 5.94·23-s + 4.18·25-s + 27-s + 5.15·29-s + 5.45·31-s − 33-s − 1.32·37-s − 6.20·39-s − 6.06·41-s − 4.79·43-s + 3.03·45-s − 2.29·47-s − 3.89·51-s − 3.97·53-s − 3.03·55-s − 4.76·57-s + 6.21·59-s − 1.35·61-s − 18.7·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.35·5-s + 0.333·9-s − 0.301·11-s − 1.71·13-s + 0.782·15-s − 0.945·17-s − 1.09·19-s − 1.23·23-s + 0.837·25-s + 0.192·27-s + 0.956·29-s + 0.979·31-s − 0.174·33-s − 0.218·37-s − 0.992·39-s − 0.947·41-s − 0.730·43-s + 0.451·45-s − 0.334·47-s − 0.545·51-s − 0.546·53-s − 0.408·55-s − 0.631·57-s + 0.808·59-s − 0.173·61-s − 2.33·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 3.03T + 5T^{2} \) |
| 13 | \( 1 + 6.20T + 13T^{2} \) |
| 17 | \( 1 + 3.89T + 17T^{2} \) |
| 19 | \( 1 + 4.76T + 19T^{2} \) |
| 23 | \( 1 + 5.94T + 23T^{2} \) |
| 29 | \( 1 - 5.15T + 29T^{2} \) |
| 31 | \( 1 - 5.45T + 31T^{2} \) |
| 37 | \( 1 + 1.32T + 37T^{2} \) |
| 41 | \( 1 + 6.06T + 41T^{2} \) |
| 43 | \( 1 + 4.79T + 43T^{2} \) |
| 47 | \( 1 + 2.29T + 47T^{2} \) |
| 53 | \( 1 + 3.97T + 53T^{2} \) |
| 59 | \( 1 - 6.21T + 59T^{2} \) |
| 61 | \( 1 + 1.35T + 61T^{2} \) |
| 67 | \( 1 - 3.15T + 67T^{2} \) |
| 71 | \( 1 - 3.94T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 6.58T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.80468627891730681439566427046, −6.66565070571352590106426577140, −6.54563152424738813601483920900, −5.45836693004466293250501782963, −4.81385442826793099148302293440, −4.12812303320685816360301325418, −2.82228459487434958252110790785, −2.32865974713272507284900857127, −1.69341489206122107345994260692, 0,
1.69341489206122107345994260692, 2.32865974713272507284900857127, 2.82228459487434958252110790785, 4.12812303320685816360301325418, 4.81385442826793099148302293440, 5.45836693004466293250501782963, 6.54563152424738813601483920900, 6.66565070571352590106426577140, 7.80468627891730681439566427046
