| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s + 2.07i·7-s − i·8-s − 9-s + 0.520·11-s − i·12-s + 2.18i·13-s − 2.07·14-s + 16-s − 2.69i·17-s − i·18-s + 6.87·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.785i·7-s − 0.353i·8-s − 0.333·9-s + 0.156·11-s − 0.288i·12-s + 0.606i·13-s − 0.555·14-s + 0.250·16-s − 0.653i·17-s − 0.235i·18-s + 1.57·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.499354156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.499354156\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.07iT - 7T^{2} \) |
| 11 | \( 1 - 0.520T + 11T^{2} \) |
| 13 | \( 1 - 2.18iT - 13T^{2} \) |
| 17 | \( 1 + 2.69iT - 17T^{2} \) |
| 19 | \( 1 - 6.87T + 19T^{2} \) |
| 23 | \( 1 - 3.86iT - 23T^{2} \) |
| 29 | \( 1 - 9.03T + 29T^{2} \) |
| 31 | \( 1 + 7.40T + 31T^{2} \) |
| 37 | \( 1 - 5.30iT - 37T^{2} \) |
| 41 | \( 1 + 3.22T + 41T^{2} \) |
| 43 | \( 1 - 9.53iT - 43T^{2} \) |
| 47 | \( 1 - 9.26iT - 47T^{2} \) |
| 53 | \( 1 + 2.43iT - 53T^{2} \) |
| 59 | \( 1 + 8.64T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 6.69iT - 67T^{2} \) |
| 71 | \( 1 + 8.16T + 71T^{2} \) |
| 73 | \( 1 - 3.84iT - 73T^{2} \) |
| 79 | \( 1 + 3.51T + 79T^{2} \) |
| 83 | \( 1 + 13.1iT - 83T^{2} \) |
| 89 | \( 1 - 9.85T + 89T^{2} \) |
| 97 | \( 1 - 10.7iT - 97T^{2} \) |
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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
−8.981223444256516473384543035679, −8.126936117890737270329944175313, −7.43046171198453703888217918476, −6.61771320524048058039159158888, −5.85194148772894204673921867922, −5.13359465128454508352171205563, −4.58675415997587091749589496546, −3.48228565079871217018865642055, −2.75148201059821619079245746232, −1.29210582847615368107349745602,
0.49371452366259607041212217149, 1.36729290703454103898909745399, 2.45852484345724089569233403713, 3.39440369989828174264294752407, 4.06798787926231589648614091685, 5.14067040815722750027142394484, 5.77967409574152406502883147446, 6.87932842397763070792390285412, 7.35890181350354425454799578950, 8.240487888080244136205380360148
