| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 5·5-s − 6·6-s − 8·8-s + 9·9-s + 10·10-s − 44·11-s + 12·12-s − 54·13-s − 15·15-s + 16·16-s − 98·17-s − 18·18-s + 60·19-s − 20·20-s + 88·22-s − 144·23-s − 24·24-s + 25·25-s + 108·26-s + 27·27-s − 210·29-s + 30·30-s + 208·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s − 1.15·13-s − 0.258·15-s + 1/4·16-s − 1.39·17-s − 0.235·18-s + 0.724·19-s − 0.223·20-s + 0.852·22-s − 1.30·23-s − 0.204·24-s + 1/5·25-s + 0.814·26-s + 0.192·27-s − 1.34·29-s + 0.182·30-s + 1.20·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9210156928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9210156928\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 98 T + p^{3} T^{2} \) |
| 19 | \( 1 - 60 T + p^{3} T^{2} \) |
| 23 | \( 1 + 144 T + p^{3} T^{2} \) |
| 29 | \( 1 + 210 T + p^{3} T^{2} \) |
| 31 | \( 1 - 208 T + p^{3} T^{2} \) |
| 37 | \( 1 + 226 T + p^{3} T^{2} \) |
| 41 | \( 1 - 502 T + p^{3} T^{2} \) |
| 43 | \( 1 - 484 T + p^{3} T^{2} \) |
| 47 | \( 1 - 232 T + p^{3} T^{2} \) |
| 53 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 764 T + p^{3} T^{2} \) |
| 61 | \( 1 + 814 T + p^{3} T^{2} \) |
| 67 | \( 1 - 60 T + p^{3} T^{2} \) |
| 71 | \( 1 - 848 T + p^{3} T^{2} \) |
| 73 | \( 1 - 958 T + p^{3} T^{2} \) |
| 79 | \( 1 + 152 T + p^{3} T^{2} \) |
| 83 | \( 1 + 308 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1094 T + p^{3} T^{2} \) |
| 97 | \( 1 + 554 T + p^{3} T^{2} \) |
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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
−9.223481185596320917289109390679, −8.213413759807812520509082356287, −7.68013402520234583450615633266, −7.11951297435883702229188616771, −5.96784404481864461309060190317, −4.90098204707229845986641835630, −3.94955470979096686056853781170, −2.68736865585840728415885485329, −2.13013436829939489891286562380, −0.47869672808352105693890506998,
0.47869672808352105693890506998, 2.13013436829939489891286562380, 2.68736865585840728415885485329, 3.94955470979096686056853781170, 4.90098204707229845986641835630, 5.96784404481864461309060190317, 7.11951297435883702229188616771, 7.68013402520234583450615633266, 8.213413759807812520509082356287, 9.223481185596320917289109390679
