| L(s) = 1 | + (1 + i)2-s + i·3-s + 2i·4-s + (−1 + i)6-s + 2·7-s + (−2 + 2i)8-s − 9-s − 2·12-s + 4i·13-s + (2 + 2i)14-s − 4·16-s + 2·17-s + (−1 − i)18-s + 4i·19-s + 2i·21-s + ⋯ |
| L(s) = 1 | + (0.707 + 0.707i)2-s + 0.577i·3-s + i·4-s + (−0.408 + 0.408i)6-s + 0.755·7-s + (−0.707 + 0.707i)8-s − 0.333·9-s − 0.577·12-s + 1.10i·13-s + (0.534 + 0.534i)14-s − 16-s + 0.485·17-s + (−0.235 − 0.235i)18-s + 0.917i·19-s + 0.436i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.809920 + 1.95532i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.809920 + 1.95532i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−11.23500814641734376313391646622, −10.06659894742228226185856138012, −9.101765155209523662115343299040, −8.186683173977034577071938201466, −7.48275774063026117609266718638, −6.27234888394639902856459079945, −5.48927151613527267240312624642, −4.41710692676585715941599052066, −3.79255416557036208511651032831, −2.22461477781248693175524155381,
1.00492182132539981994304661685, 2.34544413746819339102586438578, 3.44211313188152667594043225159, 4.78878848102329281462321447025, 5.53490213627277680282315924765, 6.56558913896775867155940768459, 7.66074180264381018851565734782, 8.563930866421553393474533315776, 9.679767950589030398625343386252, 10.65287304210731381463935127440
