| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 3.41·11-s + 12-s + 16-s − 1.41·17-s − 18-s − 2.82·19-s − 3.41·22-s + 0.828·23-s − 24-s + 27-s − 0.242·29-s + 9.07·31-s − 32-s + 3.41·33-s + 1.41·34-s + 36-s − 1.41·37-s + 2.82·38-s + 3.17·41-s − 7.41·43-s + 3.41·44-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 + 1.02·11-s + 0.288·12-s + 0.250·16-s − 0.342·17-s − 0.235·18-s − 0.648·19-s − 0.727·22-s + 0.172·23-s − 0.204·24-s + 0.192·27-s − 0.0450·29-s + 1.62·31-s − 0.176·32-s + 0.594·33-s + 0.242·34-s + 0.166·36-s − 0.232·37-s + 0.458·38-s + 0.495·41-s − 1.13·43-s + 0.514·44-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
| \(L(1)\) |
\(\approx\) |
\(1.929161843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.929161843\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 0.828T + 23T^{2} \) |
| 29 | \( 1 + 0.242T + 29T^{2} \) |
| 31 | \( 1 - 9.07T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 + 7.41T + 43T^{2} \) |
| 47 | \( 1 - 5.07T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 0.343T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 + 3.65T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 3.17T + 97T^{2} \) |
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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
−8.149164359812199010266207818103, −7.24950517140636124763302705723, −6.62584593013769516534851750280, −6.15101111722466067793954562460, −5.01098011747663744605837871863, −4.20589549938024536226897285109, −3.45805439931883024587321033668, −2.55196684681148267715812394237, −1.76058877809295394481973241293, −0.77434325714067191617028021414,
0.77434325714067191617028021414, 1.76058877809295394481973241293, 2.55196684681148267715812394237, 3.45805439931883024587321033668, 4.20589549938024536226897285109, 5.01098011747663744605837871863, 6.15101111722466067793954562460, 6.62584593013769516534851750280, 7.24950517140636124763302705723, 8.149164359812199010266207818103
