| L(s) = 1 | − i·3-s − 0.595i·7-s − 9-s + 3.35·11-s − 4.76i·13-s + 7.47i·17-s − 6.18·19-s − 0.595·21-s + 4.40i·23-s + i·27-s + 2.76·29-s + 4.48·31-s − 3.35i·33-s − 1.30i·37-s − 4.76·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.225i·7-s − 0.333·9-s + 1.01·11-s − 1.32i·13-s + 1.81i·17-s − 1.41·19-s − 0.130·21-s + 0.919i·23-s + 0.192i·27-s + 0.512·29-s + 0.806·31-s − 0.584i·33-s − 0.213i·37-s − 0.763·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.780992898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780992898\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 0.595iT - 7T^{2} \) |
| 11 | \( 1 - 3.35T + 11T^{2} \) |
| 13 | \( 1 + 4.76iT - 13T^{2} \) |
| 17 | \( 1 - 7.47iT - 17T^{2} \) |
| 19 | \( 1 + 6.18T + 19T^{2} \) |
| 23 | \( 1 - 4.40iT - 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 31 | \( 1 - 4.48T + 31T^{2} \) |
| 37 | \( 1 + 1.30iT - 37T^{2} \) |
| 41 | \( 1 + 9.40T + 41T^{2} \) |
| 43 | \( 1 + 7.59iT - 43T^{2} \) |
| 47 | \( 1 + 4.40iT - 47T^{2} \) |
| 53 | \( 1 + 8.20iT - 53T^{2} \) |
| 59 | \( 1 - 2.22T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 8.35iT - 67T^{2} \) |
| 71 | \( 1 - 6.79T + 71T^{2} \) |
| 73 | \( 1 + 7.31iT - 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 4.18iT - 83T^{2} \) |
| 89 | \( 1 + 4.25T + 89T^{2} \) |
| 97 | \( 1 + 12.2iT - 97T^{2} \) |
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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
−7.81063110387515271949451481196, −6.87298088177354555027993959906, −6.43063061451317467716059347582, −5.76228614424427126035986694985, −4.99026170187734137821570632360, −3.84818725901314100226047979879, −3.56026417451416184599465979971, −2.28622830865800989663957865380, −1.56019540997414938500333469139, −0.50511461985852276078727107685,
0.936868955346910429100289876294, 2.18573669642031733403711770580, 2.85299361466095102679821178504, 3.96169314656173944575155952057, 4.47498567203121327144233603205, 5.02768160156315078432135040878, 6.11979068029640237056532410859, 6.67228706793088870651757873876, 7.13444840626757008022527475363, 8.403625233916029106648886246363
