| L(s) = 1 | − 2-s + 1.56·3-s + 4-s − 5-s − 1.56·6-s − 4.55·7-s − 8-s − 0.544·9-s + 10-s − 11-s + 1.56·12-s − 4.63·13-s + 4.55·14-s − 1.56·15-s + 16-s + 2.48·17-s + 0.544·18-s + 0.136·19-s − 20-s − 7.13·21-s + 22-s − 6.63·23-s − 1.56·24-s + 25-s + 4.63·26-s − 5.55·27-s − 4.55·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.904·3-s + 0.5·4-s − 0.447·5-s − 0.639·6-s − 1.72·7-s − 0.353·8-s − 0.181·9-s + 0.316·10-s − 0.301·11-s + 0.452·12-s − 1.28·13-s + 1.21·14-s − 0.404·15-s + 0.250·16-s + 0.602·17-s + 0.128·18-s + 0.0313·19-s − 0.223·20-s − 1.55·21-s + 0.213·22-s − 1.38·23-s − 0.319·24-s + 0.200·25-s + 0.908·26-s − 1.06·27-s − 0.860·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6709373509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6709373509\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 + 4.55T + 7T^{2} \) |
| 13 | \( 1 + 4.63T + 13T^{2} \) |
| 17 | \( 1 - 2.48T + 17T^{2} \) |
| 19 | \( 1 - 0.136T + 19T^{2} \) |
| 23 | \( 1 + 6.63T + 23T^{2} \) |
| 29 | \( 1 - 7.05T + 29T^{2} \) |
| 31 | \( 1 - 0.167T + 31T^{2} \) |
| 37 | \( 1 + 8.45T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 47 | \( 1 - 2.16T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 0.387T + 59T^{2} \) |
| 61 | \( 1 + 6.33T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + 3.26T + 71T^{2} \) |
| 73 | \( 1 + 7.40T + 73T^{2} \) |
| 79 | \( 1 + 2.22T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 2.62T + 89T^{2} \) |
| 97 | \( 1 + 6.95T + 97T^{2} \) |
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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
−8.254017479563812785825419275181, −7.72015012258075278625328627024, −7.08675487529436005279204837106, −6.29794928118006246550657454848, −5.57948519394555243671071842948, −4.35067834166714827551969709158, −3.36648444966361004784544354713, −2.90267638650518219579133040331, −2.14709898995861867837113293022, −0.44466346430493314433545081877,
0.44466346430493314433545081877, 2.14709898995861867837113293022, 2.90267638650518219579133040331, 3.36648444966361004784544354713, 4.35067834166714827551969709158, 5.57948519394555243671071842948, 6.29794928118006246550657454848, 7.08675487529436005279204837106, 7.72015012258075278625328627024, 8.254017479563812785825419275181
