| L(s) = 1 | − 1.41i·3-s + (1 − 2i)5-s + 0.828i·7-s + 0.999·9-s + 0.828·11-s − 3.41i·13-s + (−2.82 − 1.41i)15-s − 2.82i·17-s − 19-s + 1.17·21-s + 3.65i·23-s + (−3 − 4i)25-s − 5.65i·27-s + 7.65·29-s − 1.17·31-s + ⋯ |
| L(s) = 1 | − 0.816i·3-s + (0.447 − 0.894i)5-s + 0.313i·7-s + 0.333·9-s + 0.249·11-s − 0.946i·13-s + (−0.730 − 0.365i)15-s − 0.685i·17-s − 0.229·19-s + 0.255·21-s + 0.762i·23-s + (−0.600 − 0.800i)25-s − 1.08i·27-s + 1.42·29-s − 0.210·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.835647626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.835647626\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 7 | \( 1 - 0.828iT - 7T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 + 3.41iT - 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 23 | \( 1 - 3.65iT - 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 3.41iT - 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 43 | \( 1 - 3.17iT - 43T^{2} \) |
| 47 | \( 1 + 4.82iT - 47T^{2} \) |
| 53 | \( 1 - 7.89iT - 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 - 5.65T + 61T^{2} \) |
| 67 | \( 1 + 9.89iT - 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 + 6.82iT - 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 + 4.34iT - 83T^{2} \) |
| 89 | \( 1 + 9.31T + 89T^{2} \) |
| 97 | \( 1 - 13.0iT - 97T^{2} \) |
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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
−9.144962816588938726923906498243, −8.361041696842484962960107724693, −7.65564740887684498613857379824, −6.80283812598874048057728621699, −5.92079966071911886480834971681, −5.17736046346723719750795422791, −4.24250849278742007502858327741, −2.87267544708542629005470305947, −1.76237579290334881151119298813, −0.75871766723534815344560609660,
1.60595494601426299640296757023, 2.82162875562762919341436214958, 3.91654338685982175245295465805, 4.48621100213572687884157417519, 5.62415784219016554791719470489, 6.68576595366875872954711753345, 6.98695936301981650030772179229, 8.279295101665816973640448219734, 9.058334299852890514740566738112, 10.01842312182777346271364631605
