| L(s) = 1 | + 2.07i·3-s − i·5-s + 3.88·7-s − 1.31·9-s − 4.84·11-s + 5.75·13-s + 2.07·15-s − 7.10i·17-s − 0.694·19-s + 8.07i·21-s + (2.78 + 3.90i)23-s − 25-s + 3.49i·27-s + 9.42·29-s − 5.96i·31-s + ⋯ |
| L(s) = 1 | + 1.19i·3-s − 0.447i·5-s + 1.46·7-s − 0.438·9-s − 1.46·11-s + 1.59·13-s + 0.536·15-s − 1.72i·17-s − 0.159·19-s + 1.76i·21-s + (0.581 + 0.813i)23-s − 0.200·25-s + 0.673i·27-s + 1.75·29-s − 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.153706977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.153706977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (-2.78 - 3.90i)T \) |
| good | 3 | \( 1 - 2.07iT - 3T^{2} \) |
| 7 | \( 1 - 3.88T + 7T^{2} \) |
| 11 | \( 1 + 4.84T + 11T^{2} \) |
| 13 | \( 1 - 5.75T + 13T^{2} \) |
| 17 | \( 1 + 7.10iT - 17T^{2} \) |
| 19 | \( 1 + 0.694T + 19T^{2} \) |
| 29 | \( 1 - 9.42T + 29T^{2} \) |
| 31 | \( 1 + 5.96iT - 31T^{2} \) |
| 37 | \( 1 + 1.66iT - 37T^{2} \) |
| 41 | \( 1 + 8.72T + 41T^{2} \) |
| 43 | \( 1 - 8.49T + 43T^{2} \) |
| 47 | \( 1 + 7.62iT - 47T^{2} \) |
| 53 | \( 1 + 4.40iT - 53T^{2} \) |
| 59 | \( 1 - 10.6iT - 59T^{2} \) |
| 61 | \( 1 - 3.44iT - 61T^{2} \) |
| 67 | \( 1 - 2.49T + 67T^{2} \) |
| 71 | \( 1 - 3.73iT - 71T^{2} \) |
| 73 | \( 1 - 4.45T + 73T^{2} \) |
| 79 | \( 1 + 0.694T + 79T^{2} \) |
| 83 | \( 1 - 7.34T + 83T^{2} \) |
| 89 | \( 1 + 9.40iT - 89T^{2} \) |
| 97 | \( 1 - 8.70iT - 97T^{2} \) |
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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
−9.277772687261669366960225435527, −8.589576841711827374898974663584, −7.993371028621339680418190820464, −7.13291163915227025925954132198, −5.69371761795552632690563350986, −5.05859942283008057546111279407, −4.61476716992278329233620075578, −3.62242773141936689503769759980, −2.47902109303891684354918753785, −1.03147320353219056106905923843,
1.14423162272000459806931538419, 1.91430126096061965644675236692, 2.97461053682927175628760010415, 4.27427290064454832157540534082, 5.20464332086448370285878549216, 6.21111573422536789640967973093, 6.72613043120725601069840827550, 7.86144801414986865178884741036, 8.182899303398184866236443113038, 8.651699694782136211503058555120
