| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 0.329·7-s − 8-s + 9-s − 5.03·11-s − 12-s + 0.482·13-s − 0.329·14-s + 16-s − 6.78·17-s − 18-s + 5.44·19-s − 0.329·21-s + 5.03·22-s − 6.50·23-s + 24-s − 0.482·26-s − 27-s + 0.329·28-s − 6.02·29-s + 1.31·31-s − 32-s + 5.03·33-s + 6.78·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.124·7-s − 0.353·8-s + 0.333·9-s − 1.51·11-s − 0.288·12-s + 0.133·13-s − 0.0880·14-s + 0.250·16-s − 1.64·17-s − 0.235·18-s + 1.24·19-s − 0.0718·21-s + 1.07·22-s − 1.35·23-s + 0.204·24-s − 0.0945·26-s − 0.192·27-s + 0.0622·28-s − 1.11·29-s + 0.236·31-s − 0.176·32-s + 0.876·33-s + 1.16·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6495284874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6495284874\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 0.329T + 7T^{2} \) |
| 11 | \( 1 + 5.03T + 11T^{2} \) |
| 13 | \( 1 - 0.482T + 13T^{2} \) |
| 17 | \( 1 + 6.78T + 17T^{2} \) |
| 19 | \( 1 - 5.44T + 19T^{2} \) |
| 23 | \( 1 + 6.50T + 23T^{2} \) |
| 29 | \( 1 + 6.02T + 29T^{2} \) |
| 31 | \( 1 - 1.31T + 31T^{2} \) |
| 37 | \( 1 + 0.780T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 - 4.38T + 47T^{2} \) |
| 53 | \( 1 - 1.68T + 53T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 - 4.18T + 61T^{2} \) |
| 67 | \( 1 + 3.14T + 67T^{2} \) |
| 71 | \( 1 - 5.71T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 - 3.45T + 89T^{2} \) |
| 97 | \( 1 - 9.51T + 97T^{2} \) |
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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
−8.474896901388851019683415634589, −7.67404116394994932830314090690, −7.30358696971682700861326934360, −6.24741018903407081909634358759, −5.66345051356035875747204001718, −4.85797595717638569982232629679, −3.92549016631055730658531406605, −2.69302164781324524846291613049, −1.92842968314910691022016797499, −0.51497873240470903446743274959,
0.51497873240470903446743274959, 1.92842968314910691022016797499, 2.69302164781324524846291613049, 3.92549016631055730658531406605, 4.85797595717638569982232629679, 5.66345051356035875747204001718, 6.24741018903407081909634358759, 7.30358696971682700861326934360, 7.67404116394994932830314090690, 8.474896901388851019683415634589
