| L(s) = 1 | + 0.809·2-s − 1.34·4-s + 2.87·5-s + 0.883·7-s − 2.70·8-s + 2.32·10-s + 3.13·11-s − 3.71·13-s + 0.715·14-s + 0.497·16-s − 7.58·17-s + 2.80·19-s − 3.86·20-s + 2.53·22-s − 23-s + 3.26·25-s − 3.01·26-s − 1.18·28-s + 29-s + 0.437·31-s + 5.81·32-s − 6.14·34-s + 2.53·35-s + 7.43·37-s + 2.27·38-s − 7.78·40-s + 6.99·41-s + ⋯ |
| L(s) = 1 | + 0.572·2-s − 0.672·4-s + 1.28·5-s + 0.333·7-s − 0.957·8-s + 0.735·10-s + 0.943·11-s − 1.03·13-s + 0.191·14-s + 0.124·16-s − 1.84·17-s + 0.643·19-s − 0.864·20-s + 0.540·22-s − 0.208·23-s + 0.652·25-s − 0.590·26-s − 0.224·28-s + 0.185·29-s + 0.0786·31-s + 1.02·32-s − 1.05·34-s + 0.429·35-s + 1.22·37-s + 0.368·38-s − 1.23·40-s + 1.09·41-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
| \(L(1)\) |
\(\approx\) |
\(2.734175596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.734175596\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.809T + 2T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 7 | \( 1 - 0.883T + 7T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 13 | \( 1 + 3.71T + 13T^{2} \) |
| 17 | \( 1 + 7.58T + 17T^{2} \) |
| 19 | \( 1 - 2.80T + 19T^{2} \) |
| 31 | \( 1 - 0.437T + 31T^{2} \) |
| 37 | \( 1 - 7.43T + 37T^{2} \) |
| 41 | \( 1 - 6.99T + 41T^{2} \) |
| 43 | \( 1 + 1.35T + 43T^{2} \) |
| 47 | \( 1 - 8.71T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 1.55T + 59T^{2} \) |
| 61 | \( 1 - 4.35T + 61T^{2} \) |
| 67 | \( 1 - 6.46T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 4.00T + 83T^{2} \) |
| 89 | \( 1 - 9.36T + 89T^{2} \) |
| 97 | \( 1 - 4.35T + 97T^{2} \) |
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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
−8.201889854149669613496596242194, −7.18317298754689980528908170439, −6.45816530302999160065525170134, −5.86863440047283466788983102472, −5.15804266154678537514814568970, −4.50538007256431403478063834311, −3.89606792484057860957855567489, −2.65662991455079177261077493099, −2.08070344141983009023975776450, −0.803388892812233007274242455749,
0.803388892812233007274242455749, 2.08070344141983009023975776450, 2.65662991455079177261077493099, 3.89606792484057860957855567489, 4.50538007256431403478063834311, 5.15804266154678537514814568970, 5.86863440047283466788983102472, 6.45816530302999160065525170134, 7.18317298754689980528908170439, 8.201889854149669613496596242194
