| L(s) = 1 | + 2.41·2-s + 3.82·4-s − 2·7-s + 4.41·8-s − 2.82·11-s + 4.82·13-s − 4.82·14-s + 2.99·16-s + 3.65·17-s − 1.24·19-s − 6.82·22-s + 4.41·23-s + 11.6·26-s − 7.65·28-s + 4.82·29-s + 6·31-s − 1.58·32-s + 8.82·34-s − 37-s − 3·38-s − 0.656·41-s + 1.24·43-s − 10.8·44-s + 10.6·46-s + 8.82·47-s − 3·49-s + 18.4·52-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s − 0.755·7-s + 1.56·8-s − 0.852·11-s + 1.33·13-s − 1.29·14-s + 0.749·16-s + 0.886·17-s − 0.285·19-s − 1.45·22-s + 0.920·23-s + 2.28·26-s − 1.44·28-s + 0.896·29-s + 1.07·31-s − 0.280·32-s + 1.51·34-s − 0.164·37-s − 0.486·38-s − 0.102·41-s + 0.189·43-s − 1.63·44-s + 1.57·46-s + 1.28·47-s − 0.428·49-s + 2.56·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.884980507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.884980507\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 - 4.41T + 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 41 | \( 1 + 0.656T + 41T^{2} \) |
| 43 | \( 1 - 1.24T + 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 + 2.65T + 53T^{2} \) |
| 59 | \( 1 + 7.58T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 - 7.65T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 1.65T + 83T^{2} \) |
| 89 | \( 1 - 7.17T + 89T^{2} \) |
| 97 | \( 1 + 8.48T + 97T^{2} \) |
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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
−7.60606810800422537077127505564, −6.68904641089202740308982869168, −6.29438965756007147210749776820, −5.66992732246697567231633038526, −4.97647869524672613650846063926, −4.32299153007401220445807337714, −3.35720082171793058706989011770, −3.13583402169366674658502766940, −2.19542580954023698825431035217, −0.927898651759124640029081214966,
0.927898651759124640029081214966, 2.19542580954023698825431035217, 3.13583402169366674658502766940, 3.35720082171793058706989011770, 4.32299153007401220445807337714, 4.97647869524672613650846063926, 5.66992732246697567231633038526, 6.29438965756007147210749776820, 6.68904641089202740308982869168, 7.60606810800422537077127505564
