| L(s) = 1 | + 1.94·2-s + 1.79·4-s − 3.69·5-s − 2.45·7-s − 0.390·8-s − 7.19·10-s − 4.43·11-s − 4.28·13-s − 4.78·14-s − 4.36·16-s + 1.49·17-s + 1.06·19-s − 6.64·20-s − 8.65·22-s − 23-s + 8.63·25-s − 8.35·26-s − 4.41·28-s − 29-s + 2.48·31-s − 7.71·32-s + 2.92·34-s + 9.06·35-s + 3.20·37-s + 2.08·38-s + 1.44·40-s − 4.57·41-s + ⋯ |
| L(s) = 1 | + 1.37·2-s + 0.899·4-s − 1.65·5-s − 0.927·7-s − 0.138·8-s − 2.27·10-s − 1.33·11-s − 1.18·13-s − 1.27·14-s − 1.09·16-s + 0.363·17-s + 0.244·19-s − 1.48·20-s − 1.84·22-s − 0.208·23-s + 1.72·25-s − 1.63·26-s − 0.834·28-s − 0.185·29-s + 0.445·31-s − 1.36·32-s + 0.501·34-s + 1.53·35-s + 0.527·37-s + 0.337·38-s + 0.228·40-s − 0.714·41-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
| \(L(1)\) |
\(\approx\) |
\(0.9630640357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9630640357\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.94T + 2T^{2} \) |
| 5 | \( 1 + 3.69T + 5T^{2} \) |
| 7 | \( 1 + 2.45T + 7T^{2} \) |
| 11 | \( 1 + 4.43T + 11T^{2} \) |
| 13 | \( 1 + 4.28T + 13T^{2} \) |
| 17 | \( 1 - 1.49T + 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 31 | \( 1 - 2.48T + 31T^{2} \) |
| 37 | \( 1 - 3.20T + 37T^{2} \) |
| 41 | \( 1 + 4.57T + 41T^{2} \) |
| 43 | \( 1 - 5.98T + 43T^{2} \) |
| 47 | \( 1 - 4.84T + 47T^{2} \) |
| 53 | \( 1 + 4.65T + 53T^{2} \) |
| 59 | \( 1 + 0.167T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 8.22T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 0.909T + 79T^{2} \) |
| 83 | \( 1 - 4.61T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 1.88T + 97T^{2} \) |
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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
−7.82622469294624667399988993467, −7.29218516771382425972190118913, −6.65698857659146988052005855101, −5.68551479050940859241533879181, −5.07153464522765541445679960720, −4.39497202846129659318125795809, −3.73748594930747234259691377128, −3.00672285412436408954865928796, −2.51982947433311417888570293897, −0.38540422696684188643188320941,
0.38540422696684188643188320941, 2.51982947433311417888570293897, 3.00672285412436408954865928796, 3.73748594930747234259691377128, 4.39497202846129659318125795809, 5.07153464522765541445679960720, 5.68551479050940859241533879181, 6.65698857659146988052005855101, 7.29218516771382425972190118913, 7.82622469294624667399988993467
