| L(s) = 1 | + 1.68·2-s + 0.827·4-s + 1.60·5-s − 4.81·7-s − 1.97·8-s + 2.69·10-s − 6.43·11-s + 3.23·13-s − 8.09·14-s − 4.97·16-s + 4.60·17-s + 4.55·19-s + 1.32·20-s − 10.8·22-s − 23-s − 2.43·25-s + 5.44·26-s − 3.98·28-s − 29-s − 1.11·31-s − 4.41·32-s + 7.74·34-s − 7.70·35-s + 4.51·37-s + 7.65·38-s − 3.15·40-s + 10.0·41-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 0.413·4-s + 0.715·5-s − 1.81·7-s − 0.697·8-s + 0.851·10-s − 1.94·11-s + 0.897·13-s − 2.16·14-s − 1.24·16-s + 1.11·17-s + 1.04·19-s + 0.296·20-s − 2.30·22-s − 0.208·23-s − 0.487·25-s + 1.06·26-s − 0.752·28-s − 0.185·29-s − 0.200·31-s − 0.779·32-s + 1.32·34-s − 1.30·35-s + 0.742·37-s + 1.24·38-s − 0.499·40-s + 1.57·41-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
| \(L(1)\) |
\(\approx\) |
\(2.470146086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.470146086\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.68T + 2T^{2} \) |
| 5 | \( 1 - 1.60T + 5T^{2} \) |
| 7 | \( 1 + 4.81T + 7T^{2} \) |
| 11 | \( 1 + 6.43T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 31 | \( 1 + 1.11T + 31T^{2} \) |
| 37 | \( 1 - 4.51T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 8.83T + 43T^{2} \) |
| 47 | \( 1 - 5.22T + 47T^{2} \) |
| 53 | \( 1 - 7.18T + 53T^{2} \) |
| 59 | \( 1 + 4.45T + 59T^{2} \) |
| 61 | \( 1 + 3.88T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 0.460T + 73T^{2} \) |
| 79 | \( 1 + 0.482T + 79T^{2} \) |
| 83 | \( 1 - 4.96T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 1.06T + 97T^{2} \) |
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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
−7.79027242780662287468614908451, −7.32040045955860175635784243078, −6.11550121402463024703050424425, −5.85153527322201242388287089010, −5.50503962760714068049847782930, −4.42325664457916097587039695458, −3.52009464528364823379937297861, −2.99380255354262119184160837972, −2.39712948518771264915961787827, −0.65640124901644462917629175007,
0.65640124901644462917629175007, 2.39712948518771264915961787827, 2.99380255354262119184160837972, 3.52009464528364823379937297861, 4.42325664457916097587039695458, 5.50503962760714068049847782930, 5.85153527322201242388287089010, 6.11550121402463024703050424425, 7.32040045955860175635784243078, 7.79027242780662287468614908451
