| L(s) = 1 | − 5-s + 3.37·7-s − 2.37·11-s − 3.37·13-s + 6.74·17-s + 19-s − 5.37·23-s + 25-s − 2.37·29-s + 11.1·31-s − 3.37·35-s + 6·37-s − 0.255·41-s − 4.74·43-s + 9.37·47-s + 4.37·49-s − 10.1·53-s + 2.37·55-s − 5·59-s + 12.7·61-s + 3.37·65-s − 0.744·67-s + 4.37·71-s + 14.7·73-s − 8·77-s − 2.74·79-s + 10·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.27·7-s − 0.715·11-s − 0.935·13-s + 1.63·17-s + 0.229·19-s − 1.12·23-s + 0.200·25-s − 0.440·29-s + 1.99·31-s − 0.570·35-s + 0.986·37-s − 0.0398·41-s − 0.723·43-s + 1.36·47-s + 0.624·49-s − 1.38·53-s + 0.319·55-s − 0.650·59-s + 1.63·61-s + 0.418·65-s − 0.0909·67-s + 0.518·71-s + 1.72·73-s − 0.911·77-s − 0.308·79-s + 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.875237316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.875237316\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 + 2.37T + 11T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 - 6.74T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 5.37T + 23T^{2} \) |
| 29 | \( 1 + 2.37T + 29T^{2} \) |
| 31 | \( 1 - 11.1T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 0.255T + 41T^{2} \) |
| 43 | \( 1 + 4.74T + 43T^{2} \) |
| 47 | \( 1 - 9.37T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 5T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 0.744T + 67T^{2} \) |
| 71 | \( 1 - 4.37T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 2.74T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 - 4.37T + 89T^{2} \) |
| 97 | \( 1 + 4.74T + 97T^{2} \) |
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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
−8.282241270321047635030709090503, −7.904577922833787535071223303594, −7.53381858260770277969111023531, −6.38648965430998209541156520096, −5.39933642400471517089722842931, −4.89916663875080928341708038177, −4.07192072118069796420717186976, −2.99568365351038639136920216979, −2.06133784547470349776681212413, −0.836615048177461162379251488877,
0.836615048177461162379251488877, 2.06133784547470349776681212413, 2.99568365351038639136920216979, 4.07192072118069796420717186976, 4.89916663875080928341708038177, 5.39933642400471517089722842931, 6.38648965430998209541156520096, 7.53381858260770277969111023531, 7.904577922833787535071223303594, 8.282241270321047635030709090503
