| L(s) = 1 | − 0.220·2-s + 3-s − 1.95·4-s + 3.22·5-s − 0.220·6-s − 7-s + 0.872·8-s + 9-s − 0.712·10-s + 6.18·11-s − 1.95·12-s − 0.227·13-s + 0.220·14-s + 3.22·15-s + 3.70·16-s + 2.91·17-s − 0.220·18-s + 5.03·19-s − 6.28·20-s − 21-s − 1.36·22-s + 8.84·23-s + 0.872·24-s + 5.39·25-s + 0.0502·26-s + 27-s + 1.95·28-s + ⋯ |
| L(s) = 1 | − 0.156·2-s + 0.577·3-s − 0.975·4-s + 1.44·5-s − 0.0901·6-s − 0.377·7-s + 0.308·8-s + 0.333·9-s − 0.225·10-s + 1.86·11-s − 0.563·12-s − 0.0630·13-s + 0.0590·14-s + 0.832·15-s + 0.927·16-s + 0.706·17-s − 0.0520·18-s + 1.15·19-s − 1.40·20-s − 0.218·21-s − 0.291·22-s + 1.84·23-s + 0.178·24-s + 1.07·25-s + 0.00985·26-s + 0.192·27-s + 0.368·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.803394623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.803394623\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 + 0.220T + 2T^{2} \) |
| 5 | \( 1 - 3.22T + 5T^{2} \) |
| 11 | \( 1 - 6.18T + 11T^{2} \) |
| 13 | \( 1 + 0.227T + 13T^{2} \) |
| 17 | \( 1 - 2.91T + 17T^{2} \) |
| 19 | \( 1 - 5.03T + 19T^{2} \) |
| 23 | \( 1 - 8.84T + 23T^{2} \) |
| 29 | \( 1 + 6.58T + 29T^{2} \) |
| 31 | \( 1 - 6.91T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 0.197T + 41T^{2} \) |
| 43 | \( 1 + 5.41T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 + 3.35T + 53T^{2} \) |
| 59 | \( 1 + 8.04T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 1.97T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 4.68T + 83T^{2} \) |
| 89 | \( 1 + 8.45T + 89T^{2} \) |
| 97 | \( 1 - 6.77T + 97T^{2} \) |
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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
−8.805012626744323047170774831805, −7.84290197716477826612534549643, −6.86502425941484466883498608191, −6.34460088484911413136465772743, −5.32671914982244691638092072493, −4.84600567135067351771579392703, −3.58418959871596477582936623556, −3.19111185842687391065228663689, −1.69982324444695779781797658154, −1.10575970480513337754275121732,
1.10575970480513337754275121732, 1.69982324444695779781797658154, 3.19111185842687391065228663689, 3.58418959871596477582936623556, 4.84600567135067351771579392703, 5.32671914982244691638092072493, 6.34460088484911413136465772743, 6.86502425941484466883498608191, 7.84290197716477826612534549643, 8.805012626744323047170774831805
