| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 0.979·11-s + 12-s + 0.435·13-s + 16-s − 2.79·17-s + 18-s + 7.34·19-s + 0.979·22-s − 3.34·23-s + 24-s + 0.435·26-s + 27-s + 3.74·29-s + 4.97·31-s + 32-s + 0.979·33-s − 2.79·34-s + 36-s + 4.63·37-s + 7.34·38-s + 0.435·39-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 + 0.295·11-s + 0.288·12-s + 0.120·13-s + 0.250·16-s − 0.678·17-s + 0.235·18-s + 1.68·19-s + 0.208·22-s − 0.697·23-s + 0.204·24-s + 0.0853·26-s + 0.192·27-s + 0.696·29-s + 0.894·31-s + 0.176·32-s + 0.170·33-s − 0.479·34-s + 0.166·36-s + 0.762·37-s + 1.19·38-s + 0.0696·39-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
| \(L(1)\) |
\(\approx\) |
\(4.541066255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.541066255\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.979T + 11T^{2} \) |
| 13 | \( 1 - 0.435T + 13T^{2} \) |
| 17 | \( 1 + 2.79T + 17T^{2} \) |
| 19 | \( 1 - 7.34T + 19T^{2} \) |
| 23 | \( 1 + 3.34T + 23T^{2} \) |
| 29 | \( 1 - 3.74T + 29T^{2} \) |
| 31 | \( 1 - 4.97T + 31T^{2} \) |
| 37 | \( 1 - 4.63T + 37T^{2} \) |
| 41 | \( 1 - 4.94T + 41T^{2} \) |
| 43 | \( 1 + 9.97T + 43T^{2} \) |
| 47 | \( 1 - 4.40T + 47T^{2} \) |
| 53 | \( 1 - 0.657T + 53T^{2} \) |
| 59 | \( 1 - 8.27T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 2.36T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 2.88T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.71825381578179539121602505620, −7.29269063150528196550478644364, −6.39245430884937622802799574589, −5.88574219613193094424860396553, −4.88114229137090114631395048623, −4.37782425601690433108604542064, −3.48006963483029210604499674330, −2.88851263826486144639511348755, −1.99992172742794129851399699304, −0.975955143813722509429284577011,
0.975955143813722509429284577011, 1.99992172742794129851399699304, 2.88851263826486144639511348755, 3.48006963483029210604499674330, 4.37782425601690433108604542064, 4.88114229137090114631395048623, 5.88574219613193094424860396553, 6.39245430884937622802799574589, 7.29269063150528196550478644364, 7.71825381578179539121602505620
