| L(s) = 1 | + i·2-s − 4-s + (−2.12 − 1.58i)7-s − i·8-s + 1.41i·11-s + 3.16i·13-s + (1.58 − 2.12i)14-s + 16-s − 4.47·17-s − 1.41·22-s − 6i·23-s − 3.16·26-s + (2.12 + 1.58i)28-s + 2.82i·29-s + i·32-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.801 − 0.597i)7-s − 0.353i·8-s + 0.426i·11-s + 0.877i·13-s + (0.422 − 0.566i)14-s + 0.250·16-s − 1.08·17-s − 0.301·22-s − 1.25i·23-s − 0.620·26-s + (0.400 + 0.298i)28-s + 0.525i·29-s + 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.168135423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.168135423\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.12 + 1.58i)T \) |
| good | 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 3.16iT - 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 - 9.48T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 4.47T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 9.48T + 59T^{2} \) |
| 61 | \( 1 + 13.4iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + 6.32iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 9.48T + 89T^{2} \) |
| 97 | \( 1 + 12.6iT - 97T^{2} \) |
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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
−8.727447551612241499077772055001, −7.78324298425255066745647787081, −7.04751141216354148703963462504, −6.52822501817943421484659245299, −5.91488419983971564730436024776, −4.56239991246090742060077781923, −4.35306154183261319757494533370, −3.18965029217898984918144506402, −2.04795996349006498180300292354, −0.48610428837674666607064709768,
0.842415004010476236910189537230, 2.27121932542645040762886019685, 2.95728327215938398313625027279, 3.79445072741473923276611304253, 4.70763232332643214511033296900, 5.79201271005150409892536058736, 6.10894005081573709003318072646, 7.32728838768780733427636687119, 8.020963568974225445211964758959, 9.018754323045040823342966647481
