| L(s) = 1 | − 3-s + 3.34·5-s + 2.75·7-s + 9-s + 2.38·11-s + 3.79·13-s − 3.34·15-s + 4.35·17-s − 8.55·19-s − 2.75·21-s − 8.57·23-s + 6.16·25-s − 27-s + 5.46·29-s − 9.44·31-s − 2.38·33-s + 9.19·35-s + 11.3·37-s − 3.79·39-s + 0.318·41-s + 6.22·43-s + 3.34·45-s + 10.3·47-s + 0.569·49-s − 4.35·51-s + 8.92·53-s + 7.96·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.49·5-s + 1.03·7-s + 0.333·9-s + 0.718·11-s + 1.05·13-s − 0.862·15-s + 1.05·17-s − 1.96·19-s − 0.600·21-s − 1.78·23-s + 1.23·25-s − 0.192·27-s + 1.01·29-s − 1.69·31-s − 0.414·33-s + 1.55·35-s + 1.86·37-s − 0.608·39-s + 0.0497·41-s + 0.949·43-s + 0.498·45-s + 1.51·47-s + 0.0813·49-s − 0.610·51-s + 1.22·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.914606384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.914606384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 - 3.34T + 5T^{2} \) |
| 7 | \( 1 - 2.75T + 7T^{2} \) |
| 11 | \( 1 - 2.38T + 11T^{2} \) |
| 13 | \( 1 - 3.79T + 13T^{2} \) |
| 17 | \( 1 - 4.35T + 17T^{2} \) |
| 19 | \( 1 + 8.55T + 19T^{2} \) |
| 23 | \( 1 + 8.57T + 23T^{2} \) |
| 29 | \( 1 - 5.46T + 29T^{2} \) |
| 31 | \( 1 + 9.44T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 0.318T + 41T^{2} \) |
| 43 | \( 1 - 6.22T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 8.92T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 2.18T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 9.42T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 + 0.894T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.229071249266493996159746721088, −7.26092873689132861562819712308, −6.39801105296166995691814705176, −5.74687445109503843728319763879, −5.66207062308814630124387712648, −4.33715390769587114113812842983, −3.98960485961706288458505807802, −2.42544203466916456695080521862, −1.78024007900327453637788958751, −1.00614588856557401035785748015,
1.00614588856557401035785748015, 1.78024007900327453637788958751, 2.42544203466916456695080521862, 3.98960485961706288458505807802, 4.33715390769587114113812842983, 5.66207062308814630124387712648, 5.74687445109503843728319763879, 6.39801105296166995691814705176, 7.26092873689132861562819712308, 8.229071249266493996159746721088
