| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 9-s + 2·12-s + 4·13-s + 16-s − 6·17-s − 18-s − 2·19-s − 2·24-s − 5·25-s − 4·26-s − 4·27-s − 6·29-s + 4·31-s − 32-s + 6·34-s + 36-s + 2·37-s + 2·38-s + 8·39-s − 6·41-s + 8·43-s + 12·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.577·12-s + 1.10·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.408·24-s − 25-s − 0.784·26-s − 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + 0.324·38-s + 1.28·39-s − 0.937·41-s + 1.21·43-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.001977195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.001977195\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.87946529878520861319815819980, −13.15728760628296679290378410395, −11.59222775413431567626281153330, −10.57696390398788315518055643332, −9.205467104844532229657218112988, −8.624667079434487291694904422005, −7.55563465371540871894331307120, −6.12986038892916091786195943044, −3.87985229263682105648190826671, −2.25140513775369021989127014661,
2.25140513775369021989127014661, 3.87985229263682105648190826671, 6.12986038892916091786195943044, 7.55563465371540871894331307120, 8.624667079434487291694904422005, 9.205467104844532229657218112988, 10.57696390398788315518055643332, 11.59222775413431567626281153330, 13.15728760628296679290378410395, 13.87946529878520861319815819980
