| L(s) = 1 | − 2-s + 3-s + 4-s + 1.50·5-s − 6-s − 2.36·7-s − 8-s + 9-s − 1.50·10-s + 2.57·11-s + 12-s − 6.98·13-s + 2.36·14-s + 1.50·15-s + 16-s + 4.50·17-s − 18-s − 19-s + 1.50·20-s − 2.36·21-s − 2.57·22-s − 1.78·23-s − 24-s − 2.74·25-s + 6.98·26-s + 27-s − 2.36·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.671·5-s − 0.408·6-s − 0.892·7-s − 0.353·8-s + 0.333·9-s − 0.474·10-s + 0.777·11-s + 0.288·12-s − 1.93·13-s + 0.631·14-s + 0.387·15-s + 0.250·16-s + 1.09·17-s − 0.235·18-s − 0.229·19-s + 0.335·20-s − 0.515·21-s − 0.549·22-s − 0.372·23-s − 0.204·24-s − 0.548·25-s + 1.36·26-s + 0.192·27-s − 0.446·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.636365970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.636365970\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 - 1.50T + 5T^{2} \) |
| 7 | \( 1 + 2.36T + 7T^{2} \) |
| 11 | \( 1 - 2.57T + 11T^{2} \) |
| 13 | \( 1 + 6.98T + 13T^{2} \) |
| 17 | \( 1 - 4.50T + 17T^{2} \) |
| 23 | \( 1 + 1.78T + 23T^{2} \) |
| 29 | \( 1 - 3.22T + 29T^{2} \) |
| 31 | \( 1 + 6.98T + 31T^{2} \) |
| 37 | \( 1 - 5.97T + 37T^{2} \) |
| 41 | \( 1 - 1.07T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 5.66T + 47T^{2} \) |
| 59 | \( 1 - 8.00T + 59T^{2} \) |
| 61 | \( 1 - 3.73T + 61T^{2} \) |
| 67 | \( 1 + 6.43T + 67T^{2} \) |
| 71 | \( 1 - 3.07T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 9.58T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.971763494410180902705764468011, −7.47696889075150062909710997983, −6.83425279029506189241098060559, −6.06913701454418162947262928397, −5.40748591844324805503325291998, −4.30717188207565876884785352951, −3.43192566946210112740376116090, −2.55889602442745927791684266468, −1.97801118839337479288843931053, −0.70759614321334485880884015468,
0.70759614321334485880884015468, 1.97801118839337479288843931053, 2.55889602442745927791684266468, 3.43192566946210112740376116090, 4.30717188207565876884785352951, 5.40748591844324805503325291998, 6.06913701454418162947262928397, 6.83425279029506189241098060559, 7.47696889075150062909710997983, 7.971763494410180902705764468011
