L(s) = 1 | − 1.53·3-s + 5-s + 5.03·7-s − 0.633·9-s + 3.03·11-s + 4.57·13-s − 1.53·15-s − 1.07·17-s − 19-s − 7.74·21-s + 4.11·23-s + 25-s + 5.58·27-s + 1.07·29-s + 5.58·31-s − 4.66·33-s + 5.03·35-s − 0.0947·37-s − 7.03·39-s + 10.6·41-s − 5.03·43-s − 0.633·45-s − 12.2·47-s + 18.3·49-s + 1.65·51-s + 4.09·53-s + 3.03·55-s + ⋯ |
L(s) = 1 | − 0.888·3-s + 0.447·5-s + 1.90·7-s − 0.211·9-s + 0.914·11-s + 1.26·13-s − 0.397·15-s − 0.261·17-s − 0.229·19-s − 1.68·21-s + 0.857·23-s + 0.200·25-s + 1.07·27-s + 0.199·29-s + 1.00·31-s − 0.812·33-s + 0.850·35-s − 0.0155·37-s − 1.12·39-s + 1.66·41-s − 0.767·43-s − 0.0943·45-s − 1.79·47-s + 2.61·49-s + 0.231·51-s + 0.562·53-s + 0.408·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.430671424\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.430671424\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 1.53T + 3T^{2} \) |
| 7 | \( 1 - 5.03T + 7T^{2} \) |
| 11 | \( 1 - 3.03T + 11T^{2} \) |
| 13 | \( 1 - 4.57T + 13T^{2} \) |
| 17 | \( 1 + 1.07T + 17T^{2} \) |
| 23 | \( 1 - 4.11T + 23T^{2} \) |
| 29 | \( 1 - 1.07T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 + 0.0947T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 5.03T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 - 1.39T + 59T^{2} \) |
| 61 | \( 1 - 5.69T + 61T^{2} \) |
| 67 | \( 1 + 5.28T + 67T^{2} \) |
| 71 | \( 1 + 5.67T + 71T^{2} \) |
| 73 | \( 1 - 9.07T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 + 1.95T + 83T^{2} \) |
| 89 | \( 1 + 2.18T + 89T^{2} \) |
| 97 | \( 1 + 2.16T + 97T^{2} \) |
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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
−8.359865873759207794687844428414, −7.27000057270756268874407537247, −6.44303233730091986458490223244, −5.97601458405474476513089424929, −5.15711958445439702447303999326, −4.66664625724754214903362445621, −3.86529420827288088483068605510, −2.63696223414645940202475615186, −1.52543808537217893939505576419, −0.985940231531684412537707660796,
0.985940231531684412537707660796, 1.52543808537217893939505576419, 2.63696223414645940202475615186, 3.86529420827288088483068605510, 4.66664625724754214903362445621, 5.15711958445439702447303999326, 5.97601458405474476513089424929, 6.44303233730091986458490223244, 7.27000057270756268874407537247, 8.359865873759207794687844428414