| L(s) = 1 | − i·7-s − 7i·13-s − 7·19-s − 11·31-s + 10i·37-s + 13i·43-s + 6·49-s − 61-s + 11i·67-s − 10i·73-s − 4·79-s − 7·91-s + 19i·97-s − 20i·103-s − 17·109-s + ⋯ |
| L(s) = 1 | − 0.377i·7-s − 1.94i·13-s − 1.60·19-s − 1.97·31-s + 1.64i·37-s + 1.98i·43-s + 0.857·49-s − 0.128·61-s + 1.34i·67-s − 1.17i·73-s − 0.450·79-s − 0.733·91-s + 1.92i·97-s − 1.97i·103-s − 1.62·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 7iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 11T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 13iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 11iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 19iT - 97T^{2} \) |
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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
−8.087103275878125884129847023884, −7.52273780719931405974792215507, −6.60840826800168871842885432960, −5.87894629646241984454111227752, −5.13091212331263771639935746943, −4.24933971231711257608307189887, −3.36870760737264361101332246113, −2.54375652187983303285880494985, −1.28323045869123154711371912037, 0,
1.82613389862694770517333432792, 2.29710848226740558518472567724, 3.79270123574565722716594009888, 4.20337046218374792616735621786, 5.26196614668264168212654727389, 6.01232832419749081726090108331, 6.84658880717063162143496112122, 7.31097105036755972850033686739, 8.425205710051058253338993312857
