| L(s) = 1 | + 1.78·2-s + 1.18·4-s − 3.42·5-s + 3.39·7-s − 1.45·8-s − 6.10·10-s − 1.48·11-s + 1.54·13-s + 6.06·14-s − 4.96·16-s − 1.71·17-s + 5.94·19-s − 4.04·20-s − 2.65·22-s + 23-s + 6.71·25-s + 2.74·26-s + 4.01·28-s + 29-s − 2.03·31-s − 5.94·32-s − 3.05·34-s − 11.6·35-s − 2.86·37-s + 10.6·38-s + 4.98·40-s − 2.74·41-s + ⋯ |
| L(s) = 1 | + 1.26·2-s + 0.591·4-s − 1.53·5-s + 1.28·7-s − 0.515·8-s − 1.93·10-s − 0.448·11-s + 0.427·13-s + 1.62·14-s − 1.24·16-s − 0.415·17-s + 1.36·19-s − 0.905·20-s − 0.566·22-s + 0.208·23-s + 1.34·25-s + 0.539·26-s + 0.759·28-s + 0.185·29-s − 0.364·31-s − 1.05·32-s − 0.523·34-s − 1.96·35-s − 0.471·37-s + 1.71·38-s + 0.788·40-s − 0.429·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.832884300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.832884300\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.78T + 2T^{2} \) |
| 5 | \( 1 + 3.42T + 5T^{2} \) |
| 7 | \( 1 - 3.39T + 7T^{2} \) |
| 11 | \( 1 + 1.48T + 11T^{2} \) |
| 13 | \( 1 - 1.54T + 13T^{2} \) |
| 17 | \( 1 + 1.71T + 17T^{2} \) |
| 19 | \( 1 - 5.94T + 19T^{2} \) |
| 31 | \( 1 + 2.03T + 31T^{2} \) |
| 37 | \( 1 + 2.86T + 37T^{2} \) |
| 41 | \( 1 + 2.74T + 41T^{2} \) |
| 43 | \( 1 - 2.40T + 43T^{2} \) |
| 47 | \( 1 + 0.626T + 47T^{2} \) |
| 53 | \( 1 - 1.96T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 6.34T + 61T^{2} \) |
| 67 | \( 1 + 3.34T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 8.02T + 73T^{2} \) |
| 79 | \( 1 - 2.84T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 6.93T + 97T^{2} \) |
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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
−7.894673663027216706657336405401, −7.43624486528737612855141032180, −6.62901305519467069946665423834, −5.61147790224162229719131361045, −4.95535633552528025758724930237, −4.56072468600942367506935166020, −3.69193632048184900315433572202, −3.25325746893713628748997402402, −2.11215597024898760188315926144, −0.72924300520451592490539134939,
0.72924300520451592490539134939, 2.11215597024898760188315926144, 3.25325746893713628748997402402, 3.69193632048184900315433572202, 4.56072468600942367506935166020, 4.95535633552528025758724930237, 5.61147790224162229719131361045, 6.62901305519467069946665423834, 7.43624486528737612855141032180, 7.894673663027216706657336405401
