L(s) = 1 | − 2.40i·2-s + 1.72i·3-s − 3.78·4-s − 3.36i·5-s + 4.14·6-s + (−2.55 + 0.672i)7-s + 4.28i·8-s + 0.0361·9-s − 8.09·10-s − 6.51i·12-s − 2.55·13-s + (1.61 + 6.15i)14-s + 5.79·15-s + 2.74·16-s − 3.95·17-s − 0.0869i·18-s + ⋯ |
L(s) = 1 | − 1.70i·2-s + 0.993i·3-s − 1.89·4-s − 1.50i·5-s + 1.69·6-s + (−0.967 + 0.254i)7-s + 1.51i·8-s + 0.0120·9-s − 2.56·10-s − 1.88i·12-s − 0.709·13-s + (0.432 + 1.64i)14-s + 1.49·15-s + 0.686·16-s − 0.960·17-s − 0.0204i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.118307 + 0.104394i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.118307 + 0.104394i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.55 - 0.672i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.40iT - 2T^{2} \) |
| 3 | \( 1 - 1.72iT - 3T^{2} \) |
| 5 | \( 1 + 3.36iT - 5T^{2} \) |
| 13 | \( 1 + 2.55T + 13T^{2} \) |
| 17 | \( 1 + 3.95T + 17T^{2} \) |
| 19 | \( 1 + 0.330T + 19T^{2} \) |
| 23 | \( 1 - 2.85T + 23T^{2} \) |
| 29 | \( 1 - 2.49iT - 29T^{2} \) |
| 31 | \( 1 - 0.222iT - 31T^{2} \) |
| 37 | \( 1 + 8.38T + 37T^{2} \) |
| 41 | \( 1 + 9.53T + 41T^{2} \) |
| 43 | \( 1 + 1.73iT - 43T^{2} \) |
| 47 | \( 1 - 1.23iT - 47T^{2} \) |
| 53 | \( 1 - 0.302T + 53T^{2} \) |
| 59 | \( 1 - 9.03iT - 59T^{2} \) |
| 61 | \( 1 + 5.68T + 61T^{2} \) |
| 67 | \( 1 + 6.57T + 67T^{2} \) |
| 71 | \( 1 - 5.35T + 71T^{2} \) |
| 73 | \( 1 - 6.99T + 73T^{2} \) |
| 79 | \( 1 - 2.95iT - 79T^{2} \) |
| 83 | \( 1 + 7.89T + 83T^{2} \) |
| 89 | \( 1 - 3.66iT - 89T^{2} \) |
| 97 | \( 1 + 13.6iT - 97T^{2} \) |
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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
−9.743186893366732443105682850090, −8.957329890348617112114084587081, −8.717576826598186984818312474719, −6.95569169934955170161405687152, −5.33358621471693768126761900312, −4.70735538118063159842568371615, −3.96250629566915897910851967781, −2.98655138155814784237112496070, −1.63154602470530160335173253741, −0.07495385985904185218509184933,
2.37982848520658975101731931331, 3.65460785457946008758316032457, 4.99039779152129130840188289449, 6.36587074864555309695055429047, 6.56237515690490352005172682349, 7.19706595273195251714909819133, 7.74439230948159147319170855501, 8.865136964495897931214591395674, 9.835123061186037163790208879491, 10.60005759129741134563033177658