L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 8-s + 9-s + 2·10-s − 12-s − 2·13-s − 2·15-s + 16-s + 2·17-s + 18-s − 8·19-s + 2·20-s − 23-s − 24-s − 25-s − 2·26-s − 27-s − 2·29-s − 2·30-s − 8·31-s + 32-s + 2·34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.554·13-s − 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 1.83·19-s + 0.447·20-s − 0.208·23-s − 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.371·29-s − 0.365·30-s − 1.43·31-s + 0.176·32-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.487730768\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.487730768\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.99091917601195955204136966309, −12.47239235999888074523687731605, −11.18195471039845683279157830959, −10.34314478687064358780223455819, −9.229114075576505473092106150497, −7.56963733345572718426271943739, −6.29398123690432140072875572952, −5.50969858567909471909928316815, −4.17219923557025010086906698339, −2.18807119288166115407408001809,
2.18807119288166115407408001809, 4.17219923557025010086906698339, 5.50969858567909471909928316815, 6.29398123690432140072875572952, 7.56963733345572718426271943739, 9.229114075576505473092106150497, 10.34314478687064358780223455819, 11.18195471039845683279157830959, 12.47239235999888074523687731605, 12.99091917601195955204136966309