| L(s) = 1 | − 1.41·3-s − 0.828·7-s − 0.999·9-s − 0.828·11-s + 3.41·13-s − 2.82·17-s − 19-s + 1.17·21-s + 3.65·23-s + 5.65·27-s − 7.65·29-s + 1.17·31-s + 1.17·33-s − 3.41·37-s − 4.82·39-s − 4.82·41-s + 3.17·43-s + 4.82·47-s − 6.31·49-s + 4.00·51-s − 7.89·53-s + 1.41·57-s − 1.17·59-s + 5.65·61-s + 0.828·63-s + 9.89·67-s − 5.17·69-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 0.313·7-s − 0.333·9-s − 0.249·11-s + 0.946·13-s − 0.685·17-s − 0.229·19-s + 0.255·21-s + 0.762·23-s + 1.08·27-s − 1.42·29-s + 0.210·31-s + 0.203·33-s − 0.561·37-s − 0.773·39-s − 0.754·41-s + 0.483·43-s + 0.704·47-s − 0.901·49-s + 0.560·51-s − 1.08·53-s + 0.187·57-s − 0.152·59-s + 0.724·61-s + 0.104·63-s + 1.20·67-s − 0.622·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9617753079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9617753079\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 + 7.89T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 - 5.65T + 61T^{2} \) |
| 67 | \( 1 - 9.89T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 - 6.82T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 + 4.34T + 83T^{2} \) |
| 89 | \( 1 - 9.31T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−8.599695084661606637731054770031, −7.74656679353531021670455371625, −6.78000551289898388971381236249, −6.30727749883512906252452232986, −5.52571038220754506875514793610, −4.90927037695393586447708210449, −3.87608141539292505504820917889, −3.05878823464104863268574838508, −1.90708294084359614968259958260, −0.58048344731254869550521568010,
0.58048344731254869550521568010, 1.90708294084359614968259958260, 3.05878823464104863268574838508, 3.87608141539292505504820917889, 4.90927037695393586447708210449, 5.52571038220754506875514793610, 6.30727749883512906252452232986, 6.78000551289898388971381236249, 7.74656679353531021670455371625, 8.599695084661606637731054770031
