| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 4.78·11-s + 12-s − 3.17·13-s + 16-s + 5.22·17-s − 18-s + 3.17·19-s + 4.78·22-s − 7.17·23-s − 24-s + 3.17·26-s + 27-s + 2.38·29-s − 4.17·31-s − 32-s − 4.78·33-s − 5.22·34-s + 36-s − 7.17·37-s − 3.17·38-s − 3.17·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 1.44·11-s + 0.288·12-s − 0.879·13-s + 0.250·16-s + 1.26·17-s − 0.235·18-s + 0.727·19-s + 1.02·22-s − 1.49·23-s − 0.204·24-s + 0.621·26-s + 0.192·27-s + 0.443·29-s − 0.749·31-s − 0.176·32-s − 0.832·33-s − 0.896·34-s + 0.166·36-s − 1.17·37-s − 0.514·38-s − 0.507·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.363489006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.363489006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4.78T + 11T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 - 3.17T + 19T^{2} \) |
| 23 | \( 1 + 7.17T + 23T^{2} \) |
| 29 | \( 1 - 2.38T + 29T^{2} \) |
| 31 | \( 1 + 4.17T + 31T^{2} \) |
| 37 | \( 1 + 7.17T + 37T^{2} \) |
| 41 | \( 1 - 2.05T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 8.11T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 7.11T + 61T^{2} \) |
| 67 | \( 1 + 9.56T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 3.39T + 83T^{2} \) |
| 89 | \( 1 + 9.56T + 89T^{2} \) |
| 97 | \( 1 + 1.27T + 97T^{2} \) |
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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
−7.73761204067004403637884963566, −7.61373388141640947120005598254, −6.82647068495293191803155177855, −5.60665987291757014024858991187, −5.38500209979439905436928723043, −4.21910078311497147014704510197, −3.30433028413722845728617021484, −2.58696528954720642865401641317, −1.88795351917930374039019115417, −0.61538996482536634576681037070,
0.61538996482536634576681037070, 1.88795351917930374039019115417, 2.58696528954720642865401641317, 3.30433028413722845728617021484, 4.21910078311497147014704510197, 5.38500209979439905436928723043, 5.60665987291757014024858991187, 6.82647068495293191803155177855, 7.61373388141640947120005598254, 7.73761204067004403637884963566
