| L(s) = 1 | − 0.455·3-s + 2.37·5-s + 1.84·7-s − 2.79·9-s + 3.06·11-s + 0.393·13-s − 1.08·15-s + 1.07·17-s + 7.88·19-s − 0.841·21-s − 7.82·23-s + 0.644·25-s + 2.63·27-s + 3.82·29-s + 10.4·31-s − 1.39·33-s + 4.38·35-s − 1.08·37-s − 0.179·39-s − 9.35·41-s + 8.99·43-s − 6.63·45-s − 9.78·47-s − 3.58·49-s − 0.487·51-s − 5.33·53-s + 7.27·55-s + ⋯ |
| L(s) = 1 | − 0.262·3-s + 1.06·5-s + 0.698·7-s − 0.930·9-s + 0.923·11-s + 0.109·13-s − 0.279·15-s + 0.259·17-s + 1.80·19-s − 0.183·21-s − 1.63·23-s + 0.128·25-s + 0.507·27-s + 0.711·29-s + 1.86·31-s − 0.242·33-s + 0.741·35-s − 0.178·37-s − 0.0286·39-s − 1.46·41-s + 1.37·43-s − 0.989·45-s − 1.42·47-s − 0.512·49-s − 0.0683·51-s − 0.733·53-s + 0.981·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.472761903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.472761903\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 0.455T + 3T^{2} \) |
| 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 - 1.84T + 7T^{2} \) |
| 11 | \( 1 - 3.06T + 11T^{2} \) |
| 13 | \( 1 - 0.393T + 13T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 19 | \( 1 - 7.88T + 19T^{2} \) |
| 23 | \( 1 + 7.82T + 23T^{2} \) |
| 29 | \( 1 - 3.82T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 1.08T + 37T^{2} \) |
| 41 | \( 1 + 9.35T + 41T^{2} \) |
| 43 | \( 1 - 8.99T + 43T^{2} \) |
| 47 | \( 1 + 9.78T + 47T^{2} \) |
| 53 | \( 1 + 5.33T + 53T^{2} \) |
| 59 | \( 1 - 7.04T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 1.48T + 67T^{2} \) |
| 71 | \( 1 - 9.11T + 71T^{2} \) |
| 73 | \( 1 - 4.29T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 2.37T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 1.31T + 97T^{2} \) |
| 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
−8.237172637724789893869110155208, −8.022080524367850481816026003395, −6.69258421557635025403580306335, −6.24891806262803322741865019352, −5.44417038879048321798447101500, −4.95598276493613466395873929039, −3.82426847865129014878875479091, −2.86834351927637832552058793477, −1.88352860916631640138276641380, −0.969623276910633075667986348908,
0.969623276910633075667986348908, 1.88352860916631640138276641380, 2.86834351927637832552058793477, 3.82426847865129014878875479091, 4.95598276493613466395873929039, 5.44417038879048321798447101500, 6.24891806262803322741865019352, 6.69258421557635025403580306335, 8.022080524367850481816026003395, 8.237172637724789893869110155208
