| L(s) = 1 | + 5.19·7-s + 23i·13-s + 36.3i·19-s − 19.0i·31-s + 26i·37-s + 22.5·43-s − 22·49-s − 121·61-s − 133.·67-s − 46i·73-s − 69.2i·79-s + 119. i·91-s − 167i·97-s − 69.2·103-s + 71·109-s + ⋯ |
| L(s) = 1 | + 0.742·7-s + 1.76i·13-s + 1.91i·19-s − 0.614i·31-s + 0.702i·37-s + 0.523·43-s − 0.448·49-s − 1.98·61-s − 1.99·67-s − 0.630i·73-s − 0.876i·79-s + 1.31i·91-s − 1.72i·97-s − 0.672·103-s + 0.651·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.296508749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.296508749\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 5.19T + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 23iT - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 36.3iT - 361T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 + 19.0iT - 961T^{2} \) |
| 37 | \( 1 - 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 - 22.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 121T + 3.72e3T^{2} \) |
| 67 | \( 1 + 133.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 46iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 69.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 + 167iT - 9.40e3T^{2} \) |
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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
−8.678364819490295188796619007075, −7.894993785661631975456588249576, −7.33833718571471451412211321768, −6.34235535362703635074751331265, −5.82704635205807478858879087335, −4.68610871634749318427789007045, −4.23446971439944496253336535297, −3.25401175323202661661392499292, −1.95829713338312577624799563374, −1.44186475928747060573490139894,
0.27341275325435767148436959744, 1.27344552246191028237738641377, 2.54593259929192016174334695987, 3.18827720620267704805318142214, 4.37053697287694312833245185544, 5.07064279665076735693144224056, 5.68299489415206327803483449393, 6.64823421378768092564085315962, 7.49339034403257219142822243254, 7.987008508042813177847640854353
