| L(s) = 1 | + (−1 − i)2-s + i·4-s + (0.707 − 0.707i)7-s + 1.41i·11-s − 1.41·14-s + 16-s + (1.41 − 1.41i)22-s + (1 − i)23-s + (0.707 + 0.707i)28-s + 1.41·29-s + (−1 − i)32-s + (−1.41 + 1.41i)37-s + (1.41 + 1.41i)43-s − 1.41·44-s − 2·46-s + ⋯ |
| L(s) = 1 | + (−1 − i)2-s + i·4-s + (0.707 − 0.707i)7-s + 1.41i·11-s − 1.41·14-s + 16-s + (1.41 − 1.41i)22-s + (1 − i)23-s + (0.707 + 0.707i)28-s + 1.41·29-s + (−1 − i)32-s + (−1.41 + 1.41i)37-s + (1.41 + 1.41i)43-s − 1.41·44-s − 2·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7142859937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7142859937\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
| good | 2 | \( 1 + (1 + i)T + iT^{2} \) |
| 11 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.636885548806663486220565990487, −8.794077231985517481618958032249, −8.119792907449636191952232399477, −7.31359997318531951381727217601, −6.50480973358106846094628871607, −5.02975818321283159676006712284, −4.41524000247793279355107971024, −3.10433682603351122187581516582, −2.08633881823557875856126744968, −1.10296322625892260487460429390,
1.07939456844774536381434570160, 2.69847955662211155004317495067, 3.84322538714075061591401874689, 5.38411558901332830939081181759, 5.68872367496066232242779304205, 6.75466752926570529049040403853, 7.47604305982107593305790466940, 8.334706785895143573860104278009, 8.792907969237803486442301795670, 9.309685883571173925175218442903
